1 3 points You are given the following joint and marginal di

1. (3 points) You are given the following joint and marginal distributions for two random variables (X and Y) 20.15 0.05 0.05 0.23 XI 3 0.20 0.10 0.151 0.45 5 0.05 0.15 0.10 0.30 0.40 0.30 0.30 (a)Compute E(Y),E(Y2) and Var(). Hint: Use equation (B.24) to compute Var() (b) Compute E(X)\'s 1) and E(XY = 4). ()Are Y and X independent? Explain how you can tell whether they are independent or not

Solution

(a) E(Y)= (0.40*1)+(0.30*2)+(0.30*4)= 0.40+0.60+1.20=2.20

E(Y²)= (0.40*1²)+(0.30*2²)+(0.30*4²)=0.40+1.20+4.80=6.40

Var(Y)=6.40-(2.20)²=1.56

(b) E(X|Y=1)= 2(3/8)+ 3(1/2) + 5(1/8)= 2.875

E(X|Y=4)= 2(1/6)+3(1/2)+5(1/3) = 3.50 (round upto two decimal places)

(c) Y and X are not independent.

P(X=2,Y=1)=0.15

P(X=2)= 0.25

P(Y=1)=0.40

Since here P(X=2,Y=1)P(X=2) . P(Y=1) the variables are not dependent