Question

Consider a game in which a player shoots 3 free throws; if the player makes i free throws, she draws one bill at random from a bag containing i + 1 ten-dollar bills and 5 − (i + 1) one-dollar bills. Let X be the number of free throws she makes and Y be the amount of money she wins and assume that she makes free-throws with probability 1/2.

(a) Tabulate the marginal probabilities P(X = x) for x ∈ X .

(b) Tabulate the joint probabilities P(X = x, Y = y) for (x, y) ∈ X × Y.

(c) Tabulate the marginal probabilities P(Y = y) for y ∈ Y.

(d) Check whether X and Y are independent random variables.

(e) Compute EX.

(f) Compute EY .

(g) Compute Cov(X, Y ).

(h) Compute E[Y |X = 1].

Answer 1

		Y
		10	1	P(X=x)
	0	0.025	0.1	0.125
X	1	0.15	0.225	0.375
	2	0.225	0.15	0.375
	3	0.1	0.025	0.125
P(Y=y)		0.5	0.5	1

(c) P(Y=1)=P(Y=10)=0.5.

(d) X and Y are not indpendent since

(e) E(X)=0*0.125+1*0.375+2*0.375+3*0.125=1.5

(f) E(Y)=1*0.5+10*0.5=5.5

(g) E(XY)=1*10*0.15+1*1*0.225+2*10*0.225+2*1*0.15+3*10*0.1+3*1*0.025=9.6

Cov(X,Y)=E(XY)-E(X)E(Y)=9.6-1.5*5.5=1.35.

(h) E(Y|X=1)=10*(2/5)+1*(3/5)=4.6