s****n
发帖数: 1237 | 1 记得以前好像有过，但是忘了怎么做。
X,Y are iid N(0,1)
Now given X+Y=1, what is the variance of X? | m******g
发帖数: 12 | 2 Let Z = (X+Y)/sqrt(2), then X, Z is bivariate standard normal with
correlation 1/sqrt(2). So conditioning on value of Z, X has variable 1-
rho^2
= 1/2.
Actually conditioning on any value of X+Y, the conditional variance is
1/2.
By the way, the conditional mean is given by rho Z. In this case,
conditional on Z=1/sqrt(2), the mean of X is 1/sqrt(2)*1/sqrt(2) = 1/2.
In general, if X and Z are marginally standard normal, with correlation
rho,
then conditional on Z=z, X is normal with mean rho*z, and | o*p
发帖数: 77 | 3 var(x|x+y=1)=var(y|x+y=1) by symmetry
from x+y=1;
var(x|x+y=1)+var(y|x+y=1)=1;
then var(x|x+y=1)=var(y|x+y=1)=1/2
【在 s****n 的大作中提到】
: 记得以前好像有过，但是忘了怎么做。
: X,Y are iid N(0,1)
: Now given X+Y=1, what is the variance of X?
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