1. ## 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)

2. ## Re: expectation, variance, covariance

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

3. ## Re: expectation, variance, covariance

Originally Posted by girdav
$\displaystyle E(U) =E(X)-E(Y)$, $\displaystyle \mathrm{Var}(U) = E((X-Y)^2)-(E(X-Y))^2$ and $\displaystyle \mathrm{Cov}(U,V)= E(UV)-E(U)E(V)$.
I thought that above only if x and y independent ?

4. ## Re: expectation, variance, covariance

$\displaystyle E(X_1+X_2)=E(X_1)+E(X_2)$ for two integrable random variables is always true, but $\displaystyle \mathrm{Var}(X_1+X_2)=\mathrm{Var}(X_1)+\mathrm{Va r}(X_2)$ is true if $\displaystyle X_1$ and $\displaystyle X_2$ have a variance and $\displaystyle E(X_1X_2)=E(X_1)E(X_2)$. It's true when $\displaystyle X_1$ and $\displaystyle X_2$ are independent, but the last equality can be true even if $\displaystyle X_1$ and $\displaystyle X_2$ are not independent.