Let X denote a random variable taking the values 0 and 1 wit

Let X denote a random variable taking the values 0 and 1, with Pr[X = 0] = 1/4 and Pr[X = 1] = 3/4. Let Y = 1 - 2X. What is E(X)? Using the rules for expectations, what is E(Y)? What is E(Y|X = 1)? What is E(Y|X = 0)? Use the law of iterated expectations to calculate E(Y). Does you answer agree with your answer to part (a) above? What is Var(E(Y))? What is What is Var(E(Y|X))? What is Var(E(Y|X = 1))? What is Var(E(Y|X)|X)?

Solution

The pdf of X is as follows:

E(Y) = E(1-2x)

= 1-2E(X) = 1-3/2 = -1/2

---------------------------------------

b) E(Y/x=1) = E(y=-1) = 3/4(-1) = -3/4

E(Y/x=0) = E(y=1) = 1/4(1) = 1/4

E(Y) = -3/4 +1/4 = -1/2

yes both are the same

c) Var (Y) = Var(1-2x)

= 4 var(x) = 3

d) Var (Y/x=1) = -1(3/4) = -3/4

    Var(Y/x=0) = 1(1/4) = 1/4

   Var(Y/x) = -1/2

x	0	1	Total
p	1/4	3/4	1
px	0	3/4	3/4
px^2	0	3/4	3/4
Mean	3/4
Var(x)	3/4