Question

Suppose that we have a pair of random variables X and Y with joint distribution

X
2 4 6
2 0.04 0.02 0.04
Y 4 0.12 0.06 0.12
6 0.24 0.12 0.24
(a) If we have M = max(X Y) the larger of the two observations, then calculate P{M = 6}.
(b) Show that X ami Y are independent.

Answer 1

P (m=6)= 0.60

Marginal probability of y= 0.4 , 0.2 and 0.4

Marginal probability of x= 0.1 0.3 and 0.6

Explanation:

The data given is

X 2 4 6

2 0.04 0.02 0.04

y 4 0.12 0.06 0.12

6 0.24 0.12 0.24

So the maximum value M= max (X Y) is (2,6) (6,4)(6,6)

We pick the larger value for each of the two observations

P (m=6) = 0.24+ 0.12+ 0.24= 0.60

X 2 4 6 Marginal

2 0.04 0.02 0.04 0.1

y 4 0.12 0.06 0.12 0.3

6 0.24 0.12 0.24 0.6

M. 0.4 0.2 0.4 1

The total is always equal to 1 .

X and Y are independent as the product of the marginal probability gives the joint probability.

We find the marginal probability by adding the rows and columns and then check if their product equal the given joint distribution which is true.