Suppose two regular 6sided dice are rolled Let the random va

Suppose two regular 6-sided dice are rolled. Let the random variables X and Y be the mapping for the 6 faces of the dice.

a. Find the Covariance matrix cov(X,Y)

b. find the pdf p(z) if Z=XY

Solution

Given X and Y as the random variables be the mapping for the 6 faces of the dice, then

1/6

Then E(X)=1*(1/6)+2*(1/6)+3*(1/6)+4*(1/6)+5*(1/6)+6*(1/6)=3.5 & E(Y)=1*(1/6)+2*(1/6)+3*(1/6)+4*(1/6)+5*(1/6)+6*(1/6)=3.5 & E(XY)=1*(1/6)+2²*(1/6)+3²*(1/6)+4²*(1/6)+5²*(1/6)+6²*(1/6)=15.1667. Then Cov(X,Y)= E(XY)-E(X)E(Y)=(15.1667)-(3.5*3.5)=2.9167.

If X and Y are independent and given Z =XY, then P(Z)=P(X)P(Y)=(1/36), then pdf of Z is