Let X be the number of customers arriving in a given minute

Let X be the number of customers arriving in a given minute at the drive-up window of a local bank, and let Y be the number who make withdrawals. Assume that X is Poisson distributed with expected value E(X) = 3, and that the conditional expectations and variance of Y given X = x are E(Y | x) = x/2 and Var(Y | x) = (x + 1)/3.

(a) Find E(Y).

(b) Find Var(Y).

(c) Find E(XY).

Solution

Since X~Poisson(3) ,E(X)=3 and V(X)=3

a)E(Y)=E(E(Y|X)) =E(X/2)=3/2

b) Var(Y)=E[V(Y|X)]+V[E(Y|X)]=E[(X+1)/3]+V[X/2] =4/3 +3/4 =25/12

c)E(XY)=E[E(XY|X)]=E[XE(Y|X)] =E[X²/2]=1/2 *E[X²] =1/2 *(V(X)+E²(X))=3+3² /2=6