Roll a fair 4sided dice twice If Afirst roll and Bsecond rol

Roll a fair 4-sided dice twice. If A=first roll and B=second roll, then let X=the min(A,B) and Y = |A-B|.

a) Give the joint distribution

b) Find Cov(X,Y)

Solution

a) Joint Distribution:

(A,B)---->(X,Y)

(1,1)---->(1 ,0)

(1,2)---->(1,1)

(1,3)---->(1,2)

(1,4)---->(1,3)

(2,1)---->(1,1)

(2,2)---->(2,0)

(2,3)---->(2,1)

(2,4)---->(2,2)

(3,1)---->(1,2)

(3,2)---->(2,1)

(3,3)---->(3,0)

(3,4)---->(3,1)

(4,1)---->(1,3)

(4,2)---->(2,2)

(4,3)---->(3,1)

(4,4)---->(4,0)

b)

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

E(X) = 30 /16                                        ( Summation X =30)

E(Y) = 20/16                                          (Summation Y=20)

E(XY) = 30/16                                        (Summation XY =30)

Therefore, Cov (X,Y) = 30/16 - ( 30/16 * 20/16) = - 0.4687 ( Negative Correlation ) Answer