Question

This is the probability distribution between two random variables X and Y:

Y \ X	0	1	2
3	0.1	0.2	0.2
4	0.2	0.2	0.1

a) Are those variables independent?

b) What is the marginal probability of X?

c) Find E[XY]

Answer 1

solution:

Given distribution of X and Y is

Y|X	0	1	2
3	0.1	0.2	0.2
4	0.2	0.2	0.1

b) Let's find the Margina Probability density functions of X

Px(X) =

P(X=0) = P(0,3) + P(0,4) = 0.1 + 0.2 = 0.3

P(X=1) = P(1,3) + P(1,4) = 0.2 + 0.2 = 0.4

P(X=2) = P(2,3) + P(2,4) = 0.2 + 0.1 = 0.3

a) Let's find the marginal probability density function of Y at Y=3

P(Y) =

P(Y=3) = P(0,3) + P(1,3) + P(2,3) = 0.1 + 0.2 + 0.2 = 0.5

If X and Y are independent then

P(Xi,Yj) = Px(Xi) * Py(Yj)

Let X = 0 and Y = 3

P(0,3) = 0.1

Now, P(X=0) * P(Y=3) = 0.3 * 0.5

= 0.15

!= P(0,3)

Therefore, X and Y are not independent variables

c)

E[XY] =

= ( 0 * 3 * 0.1 ) + ( 0 * 4 * 0.2 ) + ( 1 * 3 * 0.2 ) + ( 1 * 4 * 0.2 ) + ( 2 * 3 * 0.2 ) + ( 2 * 4 * 0.1 )

= 0 + 0 + 0.6 + 0.8 + 1.2 + 0.8

= 3.4

Therefore, E[XY] = 3.4