Question

1. Suppose two fair dice are rolled. Let X denote the product of the values on the dice and Y denote minimum of the two dice.
   1. Find E[X] and E[Y]
   2. Find Var X and Var Y
   3. Let Z=XY. Find E[Z].
   4. Find Cov(X,Y) and Corr(X,Y)
   5. Find E[X|Y=1] and E[Y|X=1]

Answer 1

a) The E(X) = xip(xi) and E(Y) = yip(yi)

E(X) = 12.25 and E(Y)= 2.53

b) Var X = E(X^2) - [E(X)]^2 and Var Y = E(Y^2) - [E(Y)]^2

Var X = 217.84 and Var Y = 7.17

Calculations are attached in the below picture-

c) If Z = XY

Then E (Z) = 47.44

Calculations are attached in the below pic-

d) Cov(X,Y) = E(XY)-E(X)E(Y)

from the above E(Z=XY) = 47.44, E(X)= 12.25 and E(Y)= 2.53

Cov(X,Y) = 47.44-(12.25*2.53)

Cov(X,Y) = 16.44

Cor(X,Y) = Cov(x,y)/[sqrt{Var(x)*Var(Y)}]

Cor(X,Y) = 16.44/(14.76*2.68)

Cor(X,Y) = 0.42