Question

Consider 3 jointly discrete andom variables X, Y, Z whose joint PMF is given by p(x, y, z) = {((x+ + xz? + yz), x = 1,2, y = 2,3, z = 0,1 0, otherwise a) Find the constant c. b) Find Pxy(x, y) c) Find E[xy?)

Answer 1

We would be looking at the first question, all parts here as:

a) The probabilities for each combination of values of X, Y and Z here are computed as:

x	y	z	p(x, y, z)
1	2	0	c
1	2	1	4c
1	3	0	c
1	3	1	5c
2	2	0	4c
2	2	1	8c
2	3	0	4c
2	3	1	9c

Sum of all probabilities should be 1. Therefore, we have here:
c + 4c + c + 5c + 4c + 8c + 4c + 9c = 1

36c = 1

Therefore 1/36 is the required value of c here.

b) The PDF here is obtained from the above table as:

p(x = 1, y = 2) = p(x = 1, y = 2, z = 0) + p(x = 1, y = 2, z = 1) = 5c = 5/36

Similarly all other probabilities are obtained and PDF here is given as:

p(1,2) = 5/36
p(1,3) = 6/36 = 1/6
p(2,2) = 12/36 = 1/3
p(2,3) = 13/36

This is the required joint PDF for XY

c) The expected value of xy2 here is computed as:

$E(xy^2) = \sum xy^2P(X = x, Y = y)$

E(xy2) = 4*(5/36) + 9*(1/6) + 8*(1/3) + 18*(13/36) = (5/9) + (3/2) + (8/3) + (13/2) = 8 + (29/9) = (101)/9

Therefore (101)/9 is the required expected value here.