Question

X and Y are dependent random variable with E(X)=1, E(Y)=6, Var(X)=2, and Var(Y)=2 find the following:

a. E(X-6)

b. E(XY)

c. E(X+Y)

Answer 1

Given that $E(X)=1, E(Y)=6, Var(X)=2, and\space Var(Y)=2$ then, 

$a)$

$E(X-6)=E(X)-E(6)=1-6=-5,$ since the expectation of a constant is the same constant.

$b)$

To find $E(XY)$ , we shall derive it from the covariance of the random variables $X$ and $Y$

Now,

$Cov(x,y)=E(xy)-(E(x)\times E(y))$

We make $E(xy)$ subject of the formula.

So,

$E(xy)=Cov(x,y)+(E(x)\times E(y))$

Given that $E(X)=1\space and \space E(Y)=6$ then,

$E(xy)=Cov(x,y)+(1\times 6)\\ E(xy)=Cov(x,y)+6..........(1)$

Given the value of $Cov(x,y)$, we can find the value of $E(xy)$ using equation (1) above.

$c)$

$E(X+Y)=E(X)+E(Y)=1+6=7,$ this is because, expectation of the sum of the two variables is equal to the sum of the mathematical expectation of $X$ and the mathematical expectation of $Y$.