Two variables X and Yare defined as normal random variables such that X-N(200,5^20) and Y-N(180,4^2). Find the expectation and variance for the variables A and B when. A=X+Y and B=X-Y
The expectation operator is linear, so:
$\displaystyle E(\alpha X+\beta Y)=\alpha E(X)+ \beta E(Y)$
which gives the expectation of $\displaystyle A$ and $\displaystyle B$ straight off.
Now there is an equivalent result for variances but here I will be more long winded:
$\displaystyle var(A)=E[(A-E(A))^2]=E(A^2) - (E(A))^2$
you know $\displaystyle E(A)$ from the earlier part of this post, and:
$\displaystyle E(A^2)=E[(X+Y)^2]=E(X^2)+2E(XY) + E(Y^2)$
but as $\displaystyle X$ and $\displaystyle Y$ are independant $\displaystyle E(XY)=E(X)E(Y)$, and of course:
$\displaystyle E(X^2)=var(X)+(E(X))^2$
and similarly for $\displaystyle E(Y^2).$
CB