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Номер 24.4, страница 136 - гдз по алгебре 10 класс учебник Абылкасымова, Жумагулова
Авторы: Абылкасымова А. Е., Жумагулова З. А.
Тип: Учебник
Издательство: Мектеп
Год издания: 2019 - 2026
ISBN: 978-601-07-1142-6
Утверждено Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Глава 7. Случайные величины и их числовые характеристики. Параграф 24. Числовые характеристики случайной величины - номер 24.4, страница 136.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1114015 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "136" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/24-4" "field_display_title" => "24.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113586 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Случайные величины и их числовые характеристики" "field_branch_order" => "7" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Term {#1047 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "126" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1078 #items: array:1 [ 0 => App\Models\Book {#1049 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4293 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1051 #items: array:1 [ 0 => App\Models\Term {#1048 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:1 [ 0 => App\Models\Term {#1053 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1054 #items: array:1 [ 0 => App\Models\Term {#1055 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1056 #items: array:2 [ 0 => App\Models\Term {#1057 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1058 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1060 #items: array:1 [ 0 => App\Models\Term {#1061 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1062 #items: array:1 [ 0 => App\Models\Term {#1063 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1064 #items: array:1 [ 0 => App\Models\Term {#1065 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 10 класса общественно-гуманитарного направления общеобразовательных школ" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "514" "field_priority" => "2" "field_default_folder" => "/algebra_10/Abylkasymova-u/" "field_isbn" => "978-601-07-1142-6" "field_cover" => array:1 [ 0 => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1073 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1075 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #original: array:50 [ "id" => 4293 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1051} "field_class" => Illuminate\Database\Eloquent\Collection {#1052} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1054} "field_author" => Illuminate\Database\Eloquent\Collection {#1056} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1059} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1060} "field_country" => Illuminate\Database\Eloquent\Collection {#1062} "field_city" => Illuminate\Database\Eloquent\Collection {#1064} "field_series" => Illuminate\Database\Eloquent\Collection {#1066} "field_umk" => Illuminate\Database\Eloquent\Collection {#1067} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1068} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1069} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1070} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1071} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 10 класса общественно-гуманитарного направления общеобразовательных школ" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "514" "field_priority" => "2" "field_default_folder" => "/algebra_10/Abylkasymova-u/" "field_isbn" => "978-601-07-1142-6" "field_cover" => array:1 [ 0 => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1072} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1073} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1074} "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1075} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113586 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Случайные величины и их числовые характеристики" "field_branch_order" => "7" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050} "field_page_start" => "126" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1078} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1105 #items: array:2 [ 0 => App\Models\Branch {#1113 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113586 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Случайные величины и их числовые характеристики" "field_branch_order" => "7" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1114 #items: array:1 [ 0 => App\Models\Term {#1047} ] #escapeWhenCastingToString: false } "field_page_start" => "126" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1115 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1116 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113586 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Случайные величины и их числовые характеристики" "field_branch_order" => "7" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1114} "field_page_start" => "126" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1115} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1116} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1117 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113588 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Числовые характеристики случайной величины" "field_branch_order" => "24" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:1 [ 0 => App\Models\Term {#1118 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "130" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1120 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113588 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Числовые характеристики случайной величины" "field_branch_order" => "24" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1121} "field_page_start" => "130" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1120} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1119} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1102 #items: array:3 [ 0 => App\Models\Element {#1095 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1290926 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087 #items: array:1 [ 0 => App\Models\Edition {#1094 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1092 #items: array:1 [ 0 => App\Models\Term {#1093 …30} ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1090 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1092} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1090} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1114015" "type" => "task" ] "text" => "<p><strong>24.4</strong> Заполните неполный закон распределения случайной величины, заданный в виде таблицы 31:</p><p>Таблица 31</p><svg width="300" height="60"><rect x="0" y="0" width="60" height="30" stroke="black" fill="none"/><text x="30" y="20" font-size="14" text-anchor="middle" alignment-baseline="middle">$X$</text><rect x="60" y="0" width="60" height="30" stroke="black" fill="none"/><text x="90" y="20" font-size="14" text-anchor="middle" alignment-baseline="middle">3</text><rect x="120" y="0" width="60" height="30" stroke="black" fill="none"/><text x="150" y="20" font-size="14" text-anchor="middle" alignment-baseline="middle">21</text><rect x="180" y="0" width="60" height="30" stroke="black" fill="none"/><text x="210" y="20" font-size="14" text-anchor="middle" alignment-baseline="middle">30</text><rect x="240" y="0" width="60" height="30" stroke="black" fill="none"/><text x="270" y="20" font-size="14" text-anchor="middle" alignment-baseline="middle">50</text><rect x="0" y="30" width="60" height="30" stroke="black" fill="none"/><text x="30" y="50" font-size="14" text-anchor="middle" alignment-baseline="middle">$P$</text><rect x="60" y="30" width="60" height="30" stroke="black" fill="none"/><text x="90" y="50" font-size="14" text-anchor="middle" alignment-baseline="middle">0,25</text><rect x="120" y="30" width="60" height="30" stroke="black" fill="none"/><text x="150" y="50" font-size="14" text-anchor="middle" alignment-baseline="middle">?</text><rect x="180" y="30" width="60" height="30" stroke="black" fill="none"/><text x="210" y="50" font-size="14" text-anchor="middle" alignment-baseline="middle">0,25</text><rect x="240" y="30" width="60" height="30" stroke="black" fill="none"/><text x="270" y="50" font-size="14" text-anchor="middle" alignment-baseline="middle">0,25</text></svg><p>Найдите дисперсию и среднее квадратичное отклонение.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "2756" "height" => 766 "path" => "/media/algebra_10/Abylkasymova-u/0-24/4-1.webp?ts=1753263166" ] ] ] #original: array:7 [ "id" => 1290926 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087} "task" => array:2 [ "refs" => "1114015" "type" => "task" ] "text" => "<p><strong>24.4</strong> Заполните неполный закон распределения случайной величины, заданный в виде таблицы 31:</p><p>Таблица 31</p><svg width="300" height="60"><rect x="0" y="0" width="60" height="30" stroke="black" fill="none"/><text x="30" y="20" font-size="14" text-anchor="middle" alignment-baseline="middle">$X$</text><rect x="60" y="0" width="60" height="30" stroke="black" fill="none"/><text x="90" y="20" font-size="14" text-anchor="middle" alignment-baseline="middle">3</text><rect x="120" y="0" width="60" height="30" stroke="black" fill="none"/><text x="150" y="20" font-size="14" text-anchor="middle" alignment-baseline="middle">21</text><rect x="180" y="0" width="60" height="30" stroke="black" fill="none"/><text x="210" y="20" font-size="14" text-anchor="middle" alignment-baseline="middle">30</text><rect x="240" y="0" width="60" height="30" stroke="black" fill="none"/><text x="270" y="20" font-size="14" text-anchor="middle" alignment-baseline="middle">50</text><rect x="0" y="30" width="60" height="30" stroke="black" fill="none"/><text x="30" y="50" font-size="14" text-anchor="middle" alignment-baseline="middle">$P$</text><rect x="60" y="30" width="60" height="30" stroke="black" fill="none"/><text x="90" y="50" font-size="14" text-anchor="middle" alignment-baseline="middle">0,25</text><rect x="120" y="30" width="60" height="30" stroke="black" fill="none"/><text x="150" y="50" font-size="14" text-anchor="middle" alignment-baseline="middle">?</text><rect x="180" y="30" width="60" height="30" stroke="black" fill="none"/><text x="210" y="50" font-size="14" text-anchor="middle" alignment-baseline="middle">0,25</text><rect x="240" y="30" width="60" height="30" stroke="black" fill="none"/><text x="270" y="50" font-size="14" text-anchor="middle" alignment-baseline="middle">0,25</text></svg><p>Найдите дисперсию и среднее квадратичное отклонение.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "2756" "height" => 766 "path" => "/media/algebra_10/Abylkasymova-u/0-24/4-1.webp?ts=1753263166" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1089 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1291440 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ 0 => App\Models\Edition {#1086 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1082 #items: array:1 [ 0 => App\Models\Term {#1085 …30} ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1082} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1084} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1083} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1080} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1114015" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1379" "height" => 1337 "path" => "/media/algebra_10/Abylkasymova-u/1-24/4-1.webp?ts=1753263389" ] ] ] #original: array:6 [ "id" => 1291440 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1079} "task" => array:2 [ "refs" => "1114015" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1379" "height" => 1337 "path" => "/media/algebra_10/Abylkasymova-u/1-24/4-1.webp?ts=1753263389" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1081 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1549569 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Edition {#1077 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1041 #items: array:1 [ 0 => App\Models\Term {#1043 …30} ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1034 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1039 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1041} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1034} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1039} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1114015" "type" => "task" ] "text" => "<p><strong>Заполните неполный закон распределения случайной величины, заданный в виде таблицы 31</strong></p><p>Для того чтобы закон распределения дискретной случайной величины был полным, сумма всех вероятностей $p_i$ ее возможных значений $x_i$ должна быть равна единице. Это основное свойство закона распределения:$ \sum_{i=1}^{n} p_i = 1 $</p><p>В представленной таблице даны значения случайной величины $X$: 3, 21, 30, 50. Соответствующие им вероятности $P$: 0,25; ?; 0,25; 0,25. Обозначим неизвестную вероятность, соответствующую значению $X=21$, как $p_2$.</p><p>Используя свойство о сумме вероятностей, составим уравнение:$ p_1 + p_2 + p_3 + p_4 = 1 $$ 0,25 + p_2 + 0,25 + 0,25 = 1 $$ 0,75 + p_2 = 1 $$ p_2 = 1 - 0,75 $$ p_2 = 0,25 $</p><p>Таким образом, недостающая вероятность в законе распределения равна 0,25. Заполненный закон распределения выглядит следующим образом:</p><table border="1" style="border-collapse: collapse; width: 100%; max-width: 400px; text-align: center; margin: 1em 0;"><tbody><tr><td style="padding: 8px;"><em>X</em></td><td style="padding: 8px;">3</td><td style="padding: 8px;">21</td><td style="padding: 8px;">30</td><td style="padding: 8px;">50</td></tr><tr><td style="padding: 8px;"><em>P</em></td><td style="padding: 8px;">0,25</td><td style="padding: 8px;">0,25</td><td style="padding: 8px;">0,25</td><td style="padding: 8px;">0,25</td></tr></tbody></table><p><strong>Ответ:</strong> Недостающая вероятность $P(X=21)$ равна 0,25.</p><p><strong>Найдите дисперсию и среднее квадратичное отклонение</strong></p><p>Для нахождения дисперсии и среднего квадратичного отклонения необходимо последовательно вычислить математическое ожидание, дисперсию и среднее квадратичное отклонение.</p><p><strong>1.</strong> Вычисление математического ожидания $M(X)$.Математическое ожидание (среднее значение) случайной величины вычисляется по формуле: $ M(X) = \sum_{i=1}^{n} x_i p_i $.Подставляя значения из заполненной таблицы:$ M(X) = (3 \cdot 0,25) + (21 \cdot 0,25) + (30 \cdot 0,25) + (50 \cdot 0,25) $Для упрощения вычислений вынесем общий множитель 0,25 за скобки:$ M(X) = 0,25 \cdot (3 + 21 + 30 + 50) $$ M(X) = 0,25 \cdot 104 = 26 $</p><p><strong>2.</strong> Вычисление дисперсии $D(X)$.Дисперсия — это мера разброса значений случайной величины вокруг её математического ожидания. Она вычисляется по формуле: $ D(X) = M(X^2) - [M(X)]^2 $.Сначала найдем $M(X^2)$ — математическое ожидание квадрата случайной величины:$ M(X^2) = \sum_{i=1}^{n} x_i^2 p_i $$ M(X^2) = (3^2 \cdot 0,25) + (21^2 \cdot 0,25) + (30^2 \cdot 0,25) + (50^2 \cdot 0,25) $$ M(X^2) = 0,25 \cdot (9 + 441 + 900 + 2500) $$ M(X^2) = 0,25 \cdot 3850 = 962,5 $Теперь подставим найденные значения $M(X)$ и $M(X^2)$ в формулу для дисперсии:$ D(X) = 962,5 - 26^2 = 962,5 - 676 = 286,5 $</p><p><strong>3.</strong> Вычисление среднего квадратичного отклонения $\sigma(X)$.Среднее квадратичное отклонение — это корень квадратный из дисперсии. Оно показывает среднее отклонение значений случайной величины от ее среднего значения.$ \sigma(X) = \sqrt{D(X)} $$ \sigma(X) = \sqrt{286,5} \approx 16,926 $</p><p><strong>Ответ:</strong> Дисперсия $D(X) = 286,5$; среднее квадратичное отклонение $\sigma(X) = \sqrt{286,5} \approx 16,926$.</p>" ] #original: array:6 [ "id" => 1549569 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "1114015" "type" => "task" ] "text" => "<p><strong>Заполните неполный закон распределения случайной величины, заданный в виде таблицы 31</strong></p><p>Для того чтобы закон распределения дискретной случайной величины был полным, сумма всех вероятностей $p_i$ ее возможных значений $x_i$ должна быть равна единице. Это основное свойство закона распределения:$ \sum_{i=1}^{n} p_i = 1 $</p><p>В представленной таблице даны значения случайной величины $X$: 3, 21, 30, 50. Соответствующие им вероятности $P$: 0,25; ?; 0,25; 0,25. Обозначим неизвестную вероятность, соответствующую значению $X=21$, как $p_2$.</p><p>Используя свойство о сумме вероятностей, составим уравнение:$ p_1 + p_2 + p_3 + p_4 = 1 $$ 0,25 + p_2 + 0,25 + 0,25 = 1 $$ 0,75 + p_2 = 1 $$ p_2 = 1 - 0,75 $$ p_2 = 0,25 $</p><p>Таким образом, недостающая вероятность в законе распределения равна 0,25. Заполненный закон распределения выглядит следующим образом:</p><table border="1" style="border-collapse: collapse; width: 100%; max-width: 400px; text-align: center; margin: 1em 0;"><tbody><tr><td style="padding: 8px;"><em>X</em></td><td style="padding: 8px;">3</td><td style="padding: 8px;">21</td><td style="padding: 8px;">30</td><td style="padding: 8px;">50</td></tr><tr><td style="padding: 8px;"><em>P</em></td><td style="padding: 8px;">0,25</td><td style="padding: 8px;">0,25</td><td style="padding: 8px;">0,25</td><td style="padding: 8px;">0,25</td></tr></tbody></table><p><strong>Ответ:</strong> Недостающая вероятность $P(X=21)$ равна 0,25.</p><p><strong>Найдите дисперсию и среднее квадратичное отклонение</strong></p><p>Для нахождения дисперсии и среднего квадратичного отклонения необходимо последовательно вычислить математическое ожидание, дисперсию и среднее квадратичное отклонение.</p><p><strong>1.</strong> Вычисление математического ожидания $M(X)$.Математическое ожидание (среднее значение) случайной величины вычисляется по формуле: $ M(X) = \sum_{i=1}^{n} x_i p_i $.Подставляя значения из заполненной таблицы:$ M(X) = (3 \cdot 0,25) + (21 \cdot 0,25) + (30 \cdot 0,25) + (50 \cdot 0,25) $Для упрощения вычислений вынесем общий множитель 0,25 за скобки:$ M(X) = 0,25 \cdot (3 + 21 + 30 + 50) $$ M(X) = 0,25 \cdot 104 = 26 $</p><p><strong>2.</strong> Вычисление дисперсии $D(X)$.Дисперсия — это мера разброса значений случайной величины вокруг её математического ожидания. Она вычисляется по формуле: $ D(X) = M(X^2) - [M(X)]^2 $.Сначала найдем $M(X^2)$ — математическое ожидание квадрата случайной величины:$ M(X^2) = \sum_{i=1}^{n} x_i^2 p_i $$ M(X^2) = (3^2 \cdot 0,25) + (21^2 \cdot 0,25) + (30^2 \cdot 0,25) + (50^2 \cdot 0,25) $$ M(X^2) = 0,25 \cdot (9 + 441 + 900 + 2500) $$ M(X^2) = 0,25 \cdot 3850 = 962,5 $Теперь подставим найденные значения $M(X)$ и $M(X^2)$ в формулу для дисперсии:$ D(X) = 962,5 - 26^2 = 962,5 - 676 = 286,5 $</p><p><strong>3.</strong> Вычисление среднего квадратичного отклонения $\sigma(X)$.Среднее квадратичное отклонение — это корень квадратный из дисперсии. Оно показывает среднее отклонение значений случайной величины от ее среднего значения.$ \sigma(X) = \sqrt{D(X)} $$ \sigma(X) = \sqrt{286,5} \approx 16,926 $</p><p><strong>Ответ:</strong> Дисперсия $D(X) = 286,5$; среднее квадратичное отклонение $\sigma(X) = \sqrt{286,5} \approx 16,926$.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1114016" "type" => "task" ] "previous" => array:2 [ "refs" => "1114014" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1098 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\BookPage {#1042 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1098473 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "136" "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/page-136" "field_display_title" => "136" "field_folder" => "1" "field_image_name" => "136" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1040 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1103 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => 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"2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1129} "book_page" => array:2 [ "refs" => "1098473" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "136-1.jpg" "alt" => null "width" => "3125" "height" => 4601 "path" => "/media/algebra_10/Abylkasymova-u/0-1/136-1.webp?ts=1752147543" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1098474" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1098472" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1197 #items: array:7 [ 0 => App\Models\Task {#1211 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true 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"field_display_title" => "24.1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1212} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1213} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1214} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1215} "content" => Illuminate\Database\Eloquent\Collection {#1218} "next" => array:2 [ "refs" => "1114013" "type" => "task" ] "previous" => array:2 [ "refs" => "1114011" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1224} "page" => array:2 [ "refs" => "1098473" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1217 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1114013 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "136" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/24-2" "field_display_title" => "24.2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1216 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1223 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1225 #items: array:1 [ 0 => App\Models\Branch {#1113} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1221 #items: array:2 [ 0 => App\Models\Branch {#1113} 1 => App\Models\Branch {#1117} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1220 #items: array:3 [ 0 => App\Models\Element {#1241 …30} 1 => App\Models\Element {#1243 …30} 2 => App\Models\Element {#1245 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1114014" "type" => "task" ] "previous" => array:2 [ "refs" => "1114012" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1238 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098473" "type" => "book_page" ] ] #original: array:24 [ "id" => 1114013 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "136" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/24-2" "field_display_title" => "24.2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1216} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1223} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1225} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1221} "content" => Illuminate\Database\Eloquent\Collection {#1220} "next" => array:2 [ "refs" => "1114014" "type" => "task" ] "previous" => array:2 [ "refs" => "1114012" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1238} "page" => array:2 [ "refs" => "1098473" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1222 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1114014 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "136" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/24-3" "field_display_title" => "24.3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1219 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1237 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1239 #items: array:1 [ 0 => App\Models\Branch {#1113} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1235 #items: array:2 [ 0 => App\Models\Branch {#1113} 1 => App\Models\Branch {#1117} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1234 #items: array:3 [ 0 => App\Models\Element {#1255 …30} 1 => App\Models\Element {#1257 …30} 2 => App\Models\Element {#1259 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1114015" "type" => "task" ] "previous" => array:2 [ "refs" => "1114013" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1252 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098473" "type" => "book_page" ] ] #original: array:24 [ "id" => 1114014 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "136" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/24-3" "field_display_title" => "24.3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1219} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1237} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1239} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1235} "content" => Illuminate\Database\Eloquent\Collection {#1234} "next" => array:2 [ "refs" => "1114015" "type" => "task" ] "previous" => array:2 [ "refs" => "1114013" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1252} "page" => array:2 [ "refs" => "1098473" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1236 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1114015 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "136" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/24-4" "field_display_title" => "24.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1233 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1251 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1253 #items: array:1 [ 0 => App\Models\Branch {#1113} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1249 #items: array:2 [ 0 => App\Models\Branch {#1113} 1 => App\Models\Branch {#1117} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1247 #items: array:3 [ 0 => App\Models\Element {#1095} 1 => App\Models\Element {#1089} 2 => App\Models\Element {#1081} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1114016" "type" => "task" ] "previous" => array:2 [ "refs" => "1114014" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1250 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098473" "type" => "book_page" ] ] #original: array:24 [ "id" => 1114015 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "136" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/24-4" "field_display_title" => "24.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1233} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1251} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1253} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1249} "content" => Illuminate\Database\Eloquent\Collection {#1247} "next" => array:2 [ "refs" => "1114016" "type" => "task" ] "previous" => array:2 [ "refs" => "1114014" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1250} "page" => array:2 [ "refs" => "1098473" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1248 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1114016 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "136" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/24-5" "field_display_title" => "24.5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1261 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1262 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1263 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1264 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1267 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1114017" "type" => "task" ] "previous" => array:2 [ "refs" => "1114015" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1273 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098473" "type" => "book_page" ] ] #original: array:24 [ "id" => 1114016 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "136" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/24-5" "field_display_title" => "24.5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1261} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1262} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1263} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1264} "content" => Illuminate\Database\Eloquent\Collection {#1267} "next" => array:2 [ "refs" => "1114017" "type" => "task" ] "previous" => array:2 [ "refs" => "1114015" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1273} "page" => array:2 [ "refs" => "1098473" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Task {#1266 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1114017 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "136" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/24-6" "field_display_title" => "24.6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1265 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1272 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1274 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1270 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1269 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1114018" "type" => "task" ] "previous" => array:2 [ "refs" => "1114016" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1287 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098473" "type" => "book_page" ] ] #original: array:24 [ "id" => 1114017 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "136" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/24-6" "field_display_title" => "24.6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1265} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1272} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1274} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1270} "content" => Illuminate\Database\Eloquent\Collection {#1269} "next" => array:2 [ "refs" => "1114018" "type" => "task" ] "previous" => array:2 [ "refs" => "1114016" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1287} "page" => array:2 [ "refs" => "1098473" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 6 => App\Models\Task {#1271 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1114018 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "136" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/24-7" "field_display_title" => "24.7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1268 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1286 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1288 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1284 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1283 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1114019" "type" => "task" ] "previous" => array:2 [ "refs" => "1114017" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1301 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098473" "type" => "book_page" ] ] #original: array:24 [ "id" => 1114018 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "136" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/24-7" "field_display_title" => "24.7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1268} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1286} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1288} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1284} "content" => Illuminate\Database\Eloquent\Collection {#1283} "next" => array:2 [ "refs" => "1114019" "type" => "task" ] "previous" => array:2 [ "refs" => "1114017" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1301} "page" => array:2 [ "refs" => "1098473" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1098473 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "136" "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/page-136" "field_display_title" => "136" "field_folder" => "1" "field_image_name" => "136" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1040} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1103} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1112} "content" => Illuminate\Database\Eloquent\Collection {#1128} "next" => array:2 [ "refs" => "1098474" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1098472" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1197} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1114015 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "136" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/24-4" "field_display_title" => "24.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1105} "content" => Illuminate\Database\Eloquent\Collection {#1102} "next" => array:2 [ "refs" => "1114016" "type" => "task" ] "previous" => array:2 [ "refs" => "1114014" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1098} "page" => array:2 [ "refs" => "1098473" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№24.4 (с. 136)
Условие. №24.4 (с. 136)
скриншот условия
24.4 Заполните неполный закон распределения случайной величины, заданный в виде таблицы 31:
Таблица 31
Найдите дисперсию и среднее квадратичное отклонение.
Решение. №24.4 (с. 136)
Решение 2. №24.4 (с. 136)
Заполните неполный закон распределения случайной величины, заданный в виде таблицы 31
Для того чтобы закон распределения дискретной случайной величины был полным, сумма всех вероятностей $p_i$ ее возможных значений $x_i$ должна быть равна единице. Это основное свойство закона распределения:$ \sum_{i=1}^{n} p_i = 1 $
В представленной таблице даны значения случайной величины $X$: 3, 21, 30, 50. Соответствующие им вероятности $P$: 0,25; ?; 0,25; 0,25. Обозначим неизвестную вероятность, соответствующую значению $X=21$, как $p_2$.
Используя свойство о сумме вероятностей, составим уравнение:$ p_1 + p_2 + p_3 + p_4 = 1 $$ 0,25 + p_2 + 0,25 + 0,25 = 1 $$ 0,75 + p_2 = 1 $$ p_2 = 1 - 0,75 $$ p_2 = 0,25 $
Таким образом, недостающая вероятность в законе распределения равна 0,25. Заполненный закон распределения выглядит следующим образом:
| X | 3 | 21 | 30 | 50 |
| P | 0,25 | 0,25 | 0,25 | 0,25 |
Ответ: Недостающая вероятность $P(X=21)$ равна 0,25.
Найдите дисперсию и среднее квадратичное отклонение
Для нахождения дисперсии и среднего квадратичного отклонения необходимо последовательно вычислить математическое ожидание, дисперсию и среднее квадратичное отклонение.
1. Вычисление математического ожидания $M(X)$.Математическое ожидание (среднее значение) случайной величины вычисляется по формуле: $ M(X) = \sum_{i=1}^{n} x_i p_i $.Подставляя значения из заполненной таблицы:$ M(X) = (3 \cdot 0,25) + (21 \cdot 0,25) + (30 \cdot 0,25) + (50 \cdot 0,25) $Для упрощения вычислений вынесем общий множитель 0,25 за скобки:$ M(X) = 0,25 \cdot (3 + 21 + 30 + 50) $$ M(X) = 0,25 \cdot 104 = 26 $
2. Вычисление дисперсии $D(X)$.Дисперсия — это мера разброса значений случайной величины вокруг её математического ожидания. Она вычисляется по формуле: $ D(X) = M(X^2) - [M(X)]^2 $.Сначала найдем $M(X^2)$ — математическое ожидание квадрата случайной величины:$ M(X^2) = \sum_{i=1}^{n} x_i^2 p_i $$ M(X^2) = (3^2 \cdot 0,25) + (21^2 \cdot 0,25) + (30^2 \cdot 0,25) + (50^2 \cdot 0,25) $$ M(X^2) = 0,25 \cdot (9 + 441 + 900 + 2500) $$ M(X^2) = 0,25 \cdot 3850 = 962,5 $Теперь подставим найденные значения $M(X)$ и $M(X^2)$ в формулу для дисперсии:$ D(X) = 962,5 - 26^2 = 962,5 - 676 = 286,5 $
3. Вычисление среднего квадратичного отклонения $\sigma(X)$.Среднее квадратичное отклонение — это корень квадратный из дисперсии. Оно показывает среднее отклонение значений случайной величины от ее среднего значения.$ \sigma(X) = \sqrt{D(X)} $$ \sigma(X) = \sqrt{286,5} \approx 16,926 $
Ответ: Дисперсия $D(X) = 286,5$; среднее квадратичное отклонение $\sigma(X) = \sqrt{286,5} \approx 16,926$.
Другие задания:
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