Question

A fair coin is tossed three times. Let X be the number of heads among the first two tosses and Y be the number of heads among the last two tosses. What is the joint probability mass function of X and Y? What are the marginal probability mass function of X and Y i.e. p_X (x)and p_Y (y)? Find E(X) and E(Y). What is Cov(X,Y) What is Corr (X,Y) Are X and Y independent? Explain. Find the conditional probability mass functionP(X/Y=1). Find E(Y/X=1).

Answer 1

a. Joint pmf of X and Y is

b.

Marginal pmf of X and Y are

c. E(X) = {x . p_x(x) } = 0*(2/8) + 1*(4/8) + 2*(2/8) = 1

E(Y) = {y . p_Y(y) } = 0*(2/8) + 1*(4/8) + 2*(2/8) = 1

d. E(XY) =  xyp(x,y) =0*0*(1/8) + 0*1*(1/8) + 0*2*0 + 1*0*(1/8) + 1*1*(2/8) + 1*2*(1/8) + 2*0*0 +2*1*(1/8) + 2*2*(1/8) = 10/8

Cov(X,Y) = E(XY) - E(X)E(Y) = 10/8 - 1*1 = 0.25

e. Var(X) =  {x². p_x(x) } -[{x . p_x(x) } ] ² = 0*(2/8) + 1*(4/8) + 4*(2/8) - 1² = 0.5

Var(Y) =0.5

Corr(X,Y) = Cov(X,Y)/_xy = 0.25/sqrt(0.5*0.5) = 0.25/0.5 = 0.5