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Консультация № 190948
05.05.2017, 20:46
0.00 руб.
0
2
1
Здравствуйте! Прошу помощи в следующем вопросе:
Случайная величина имеет распределение Коши с плотностью
f(x) = а/(1 + x^2) , при −∞ < x < ∞.
Найти:
а) коэффициент a;
б) функцию распределения F (x);
в) моду mod(X) и медиану med(X);
г) вероятность P{X∈ [−1, 1]};
д) выяснить, существует ли EX.
Заранее спасибо!
Случайная величина имеет распределение Коши с плотностью
f(x) = а/(1 + x^2) , при −∞ < x < ∞.
Найти:
а) коэффициент a;
б) функцию распределения F (x);
в) моду mod(X) и медиану med(X);
г) вероятность P{X∈ [−1, 1]};
д) выяснить, существует ли EX.
Заранее спасибо!
Обсуждение
nikolay.kireev1
06.05.2017, 14:35
общий
это ответ
Здравствуйте, nikolay.kireev1!
а) Вычислим коэффициент
из условия нормировки плотности вероятности 
(см. здесь).
б) Выведем выражение для функции распределения
в) Вычислим моду заданного распределения:
потому что при
плотность вероятности 
достигает максимума
Вычислим медиану заданного распределения:
потому что при выполняется соотношение
то есть
г) Вычислим вероятность
д) Если
- математическое ожидание случайной величины 
то для распределения Коши оно не существует, потому что определяющий его интеграл расходится. Покажите это самостоятельно, иначе, "изучив" теорию вероятностей и математическую статистику, Вы ничему не научитесь. Если возникнут вопросы, задавайте их в мини-форуме консультации. 
а) Вычислим коэффициент
из условия нормировки плотности вероятности (см. здесь).б) Выведем выражение для функции распределения
в) Вычислим моду заданного распределения:
потому что при
плотность вероятности достигает максимумаВычислим медиану заданного распределения:
потому что при
выполняется соотношението есть
г) Вычислим вероятность
д) Если
- математическое ожидание случайной величины то для распределения Коши оно не существует, потому что определяющий его интеграл расходится. Покажите это самостоятельно, иначе, "изучив" теорию вероятностей и математическую статистику, Вы ничему не научитесь. Если возникнут вопросы, задавайте их в мини-форуме консультации.
5
Большое спасибо!
Об авторе:
Facta loquuntur.
Facta loquuntur.
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