推 bawd968 : 原來如此 太感謝了～ 10/24 17:30
※ 引述《bawd968 (bawd968)》之銘言：
: 隨機過程的題目，想了很久還是不會解，希望有人可以列出詳細的過程
: 題目如下圖
: https://i.imgur.com/oaq1k7Q.png
: 先謝謝各位～～
x_1 = x
x_2 = y
x_3 = z
f(x, y, z) = (2π√π)^(-1) exp(-x^2 - y^2 + √2 xy - (1/2)z^2)
(a)
∞ ∞
f_X = ∫ ∫f(x,y,z)dydz
-∞-∞
= (2π√π)^(-1) ∫exp(-(y - x/√2)^2 - (1/2)x^2 - (1/2)z^2) dydz
= (2π√π)^(-1) exp(-(1/2)x^2) (π)√2
= [1/√(2π)] exp(-(1/2)x^2)
f_Z = ∫f(x,y,z)dxdy
= (2π√π)^(-1) exp(-(1/2)z^2) * π√2
= [1/√(2π)] exp(-(1/2)z^2)
(b)
μ_X = ∫xf_x dx = 0 = μ_Y = μ_Z
E[XY] = (2π√π)^(-1) √(2π)∫ xyexp(-(x - y/√2)^2 - (1/2)y^2)dxdy
= 1/[π√2] [1/√2]∫y^2 exp(-(x - y/√2)^2 - (1/2)y^2)dxdy
= 1/[2π] * √π * √(2π)
= 1/√2
E[XZ] = 0 = E[YZ]
E[ZZ] = [1/√(2π)] ∫ z^2 exp(-(1/2)z^2) dz
= 1
E[XX] = (2π√π)^(-1) √(2π) √π∫x^2 exp(-(1/2)x^2) dx
= 1 = E[YY]
Cov_ij = 1 for i = j
1/√2 for (i, j) = (1, 2) or (2, 1)
0 otherwise
--
※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 61.56.10.112
※ 文章網址: https://www.ptt.cc/bbs/Math/M.1508819666.A.09A.html