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Консультация № 177669

04.04.2010, 18:01

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Образование

Помогите решить, пожалуйста.
Найти математическое ожидание, дисперсию и корреляционную функцию случайной функции X(t)=Ucos2t+V sint+t, где U и V – – некоррелированные случайные величины, причем M(u)=1, M(v)=2,D(u)=3,D(v)=4

Обсуждение

Неизвестный

04.04.2010, 23:42

общий

это ответ

Здравствуйте, resident291288.

Математическое ожидание:
m_X(t)= m_U*cos(2t)+m_V*sin(t)+m_t
m_U*cos(2t) = cos(2t)*M(U)=cos(2t)
m_V*sin(t) = sin(t)*M(V)=2*sin(t)
m_t=t
m_X(t)=cos(2t)+2*sin(t)+t

Корреляционная функция:
Т.к. t - не является случайной величиной и U,V - некоррелированы, то

K_X(t₁,t₂)=K_{U*cos(2t)+V*sin(t)}(t₁,t₂)=K_U*cos(2t)(t₁,t₂)+K_V*sin(t)(t₁,t₂)

K_U*cos(2t)(t₁,t₂)=cos(2t₁)*cos(2t₂)*D(U)=3*cos(2t₁)*cos(2t₂)
аналогично
K_V*sin(t)(t₁,t₂)=4*sin(t₁)*sin(t₂)

K_X(t₁,t₂)=3*cos(2t₁)*cos(2t₂)+4*sin(t₁)*sin(t₂)

Дисперсия:
D_X(t)=K_X(t,t)=3*cos²(2t)+4*sin²(t)

Ответ:
m_X(t)=cos(2t)+2*sin(t)+t
D_X(t)=3*cos²(2t)+4*sin²(t)
K_X(t₁,t₂)=3*cos(2t₁)*cos(2t₂)+4*sin(t₁)*sin(t₂)

5

Спасибо огромное

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