j******n
发帖数: 91 | 1 Assuming x - N(mu_x, sigma_x), y - N(mu_y, sigma_y), rho(x, y)=rho. Given z
= x+y (z is known), what is E(X|X+Y=z)? | k**y
发帖数: 920 | 2 sigma_x和sigma_y是std对吧？
let Y'=X+Y
then EY'=mu_x+mu_y
std(Y')=sqrt(varX+varY+2rho*sigma_x*sigm_y)
Cov(X,Y')=Cov(X,X+Y)=varX+Cov(X,Y)=sigma_x^2+rho*sigma_x*simga_y
根据Normal Dist条件分布的公式
E(X|Y'=z) = mu_x + sigma_x/std(Y')*corr(X,Y')*(z-EY')
= mu_x + Cov(X,Y')/std(Y')^2 *(z-mu_x-my_y)
= mu_x + (sigma_x^2+rho*sigma_x*sigma_y)/(sigma_x^2+sigma_y^2+2rho
*sigma_x*sigma_y *(z-mu_x_mu_y)
z
【在 j******n 的大作中提到】
: Assuming x - N(mu_x, sigma_x), y - N(mu_y, sigma_y), rho(x, y)=rho. Given z
: = x+y (z is known), what is E(X|X+Y=z)?
| j******n
发帖数: 91 | 3 多谢多谢
2rho
【在 k**y 的大作中提到】
: sigma_x和sigma_y是std对吧？
: let Y'=X+Y
: then EY'=mu_x+mu_y
: std(Y')=sqrt(varX+varY+2rho*sigma_x*sigm_y)
: Cov(X,Y')=Cov(X,X+Y)=varX+Cov(X,Y)=sigma_x^2+rho*sigma_x*simga_y
: 根据Normal Dist条件分布的公式
: E(X|Y'=z) = mu_x + sigma_x/std(Y')*corr(X,Y')*(z-EY')
: = mu_x + Cov(X,Y')/std(Y')^2 *(z-mu_x-my_y)
: = mu_x + (sigma_x^2+rho*sigma_x*sigma_y)/(sigma_x^2+sigma_y^2+2rho
: *sigma_x*sigma_y *(z-mu_x_mu_y)
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