Let X Y be independent random variables Y having Normal0 l d

Let X, Y be independent random variables, Y having Normal(0, l) distribution and X has distribution P(X = 1.5) = P{X = 0.5) = 1/2. Let Z = XY. Compute E(Z\\Y). E(Z2\\Y).

Solution

Y is normal with mean =0 and std dev =1

X is having pdf as given.

E(x) = 1.5(0.5) + 0.5(0.5)

= 1.00

Hence E(z) = E(xy) = 1(0) =0 (as x and y are independent0

E(Z/Y) = E(x) = 1

b) E(Z^2/Y) = E(X^2 Y) = E(X^2) E(Y)

=0