A die with three sides 123 is tossed two times Let X equal t

A die with three sides (1,2,3) is tossed two times. Let X equal themaximum of two observations and let Y equal the product of twoobservations. Find the correlation coefficient of X and Y.

Hint: construct three tables

1)for sample space for two tosses

2)a table for the values of X and Y for each outcome

3)the table for pmf

Solution

A die with three sides (1,2,3) is tossed two times.

Let X equal themaximum of two observations and

let Y equal the product of twoobservations.

Sample space = { (1,1) (1,2) (1,3) (2,1) (2,2) (2,3) (3,1) (3,2) (3,3) }

number of sample points are 9.

The possible values of X are 1,2 and 3

Now probability distribution of X is,

the possible values of Y are 1,2,3,4,6 and 9.

Joint probability distribution of X and Y is,

E(X) = x*P(x) = 1*1/9 + 2*3/9 + 3*5/9 = 2.44

E(Y) = y*P(y) = 1*1/9 + 2*2/9 + 3*2/9 + 4*1/9 + 6*2/9 + 9 * 1/9 = 4

Var(X) = (x - E(X) )² P(x)

= (1-2.44)² * 1/9 + (2-2.44)² * 3/9 + (3-2.44)² * 5/9 = 0.4691

Var(Y) =  (y - E(Y) )² P(y)

= (1-4)² * 1/9 + (2-4)² * 2/9 + (3-4)² * 2/9 + (4-4)² * 1/9 + (6-4)² * 2/9 + (9-4)² * 1/9 = 5.7777

E(XY) = xy * P(x,y)

= 1*1/9 + 2*1/9 + 3*1/3 + 2*4/9 + 18*2/9 + 27 * 1/9 = 9.2222

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

= 9.2222 - 2.44*4 = -0.5378

r = cov(X,Y) / sqrt(var(X) * var(Y) )

= -0.5378 / sqrt(0.4691*5.7777) = -0.32667