# expectation, variance, covariance

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• Jul 30th 2011, 12:47 AM
bingunginter
expectation, variance, covariance
I'm stuck on this problem :

X and Y is normal random var with mean(x) = 1 and mean(y) = 2 and variance(x)=1 and variance(y) = 4 and covariance(x,y)=1.

U = X-Y
V = X+Y

find E(U),Var(U),Cov(U,V)
• Jul 30th 2011, 12:57 AM
girdav
Re: expectation, variance, covariance
$E(U) =E(X)-E(Y)$, $\mathrm{Var}(U) = E((X-Y)^2)-(E(X-Y))^2$ and $\mathrm{Cov}(U,V)= E(UV)-E(U)E(V)$.
• Jul 30th 2011, 12:58 AM
bingunginter
Re: expectation, variance, covariance
Quote:

Originally Posted by girdav
$E(U) =E(X)-E(Y)$, $\mathrm{Var}(U) = E((X-Y)^2)-(E(X-Y))^2$ and $\mathrm{Cov}(U,V)= E(UV)-E(U)E(V)$.

I thought that above only if x and y independent ?
• Jul 30th 2011, 01:04 AM
girdav
Re: expectation, variance, covariance
$E(X_1+X_2)=E(X_1)+E(X_2)$ for two integrable random variables is always true, but $\mathrm{Var}(X_1+X_2)=\mathrm{Var}(X_1)+\mathrm{Va r}(X_2)$ is true if $X_1$ and $X_2$ have a variance and $E(X_1X_2)=E(X_1)E(X_2)$. It's true when $X_1$ and $X_2$ are independent, but the last equality can be true even if $X_1$ and $X_2$ are not independent.