Please show me how to solve problems below Please give me crystal clear solution Textbook Answers: a) 1 b) 2 Suppose that the conditional distribution of Y given X = x is Poisson with mean E(Y|x) = x,Y |x ~ POI(x) and that X ~ EXP(1) (a) Find E(Y) (b) Find Var(Y) Solution Y|X follows poisson(x) and X follows exp(1) a) E(Y) = E(E(Y|X)) = E(X) = 1 since E(Y) can always be written as E(E(Y|X)) b) E(Y^2) = E(E(Y^2/X)) = E(X + X^2) = 1+ 1 = 2 since E(Y^2/X) = var(Y/X) + E(Y/X)^2 = X+X^2.

1. Please show me how to solve problems below
Please give me crystal clear solution
Textbook Answers:
a) 1
b) 2
Suppose that the conditional distribution of Y given X = x is Poisson with mean E(Y|x) = x,Y |x
~ POI(x) and that
X ~ EXP(1)
(a) Find E(Y)
(b) Find Var(Y)
Solution
Y|X follows poisson(x) and X follows exp(1)
a) E(Y) = E(E(Y|X)) = E(X) = 1 since E(Y) can always be written as E(E(Y|X))
b) E(Y^2) = E(E(Y^2/X)) = E(X + X^2) = 1+ 1 = 2 since E(Y^2/X) = var(Y/X) + E(Y/X)^2 =
X+X^2