1. ## addition of expectation and variance

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

2. Originally Posted by pinkpoodle
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

3. sorry i do not understand would you beable to put in the values and the exact answers. many thanks

4. Actually that was great help, I understand now.