Suppose that X and Y are integer valued random variables with joint probability

Question

Suppose that X and Y are integer valued random variables with joint probability mass function given by PX, Y(a, b) = {1/4a, for 1 le b le a le 4, 0, otherwise. Show that this is indeed a joint probability mass function. Find the marginal probability mass function of X and Y. Find P(X = Y + 1).

Exercise 6.4. Suppose that Xand Y are integer valued random variables with joint probability mass function given by for 1 bs a 4, 4a px,Y (a, b) otherwise. (a) Show that this is indeed a joint probability mass function. (b) Find the marginal probability mass function of X and Y. (c) Find PCX Y+1).

Explanation / Answer

Back-up Theory

1. A pmf must satisfy two conditions:

i) it must take only non-negative values, but less than 1

ii) sum of all probabilities must be unity.

2. Marginal pmf of one variable is obtained by summing the joint probabilities over all possible values of the other variable.

Solution

Given X and Y are integer value variables and 1 b a 4, X = 1, 2, 3, 4 and Y = 1, 2, 3, 4 subject to the condition Y X.

Based on this, the joint probabilities are presented in the following table:

From the above table, it is clear that each probabilty value is positive proper fraction and sum of all probability values is unity (1).

Hence, the given probability function is indeed a pmf. Answers (a)

Marginal pmf of X is the sum of all probabilities of (x, y ) over y and Marginal pmf of Y is the sum of all probabilities of (x, y ) over X. These are given in the last column and last row respectively of the above table.Answers (b)

P(X = Y + 1) = P(X = 2, 3, 4) = 1 - P(X = 1) = 1 - 1/4 = 3/4.Answers (c)

Joint Probability y = 1 y = 2 y = 3 y = 4 Total x = 1 1/4 0 0 0 1/4 x = 2 1/8 1/8 0 0 1/4 x = 3 1/12 1/12 1/12 0 1/4 x = 4 1/16 1/16 1/16 1/16 1/4 Total 25/48 13/48 7/48 3/48 1

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