Joint and marginal probability mass functions

Joint and marginal probability mass functions

In real life situations we may be observed two or more random variables on the individuals simultaneously. For instance, blood pressure and cholesterol for each individual are measured simultaneously. In such cases we require the concept of bi-variate random variable represented by ( X, Y ), where X and Y are univariate random variables.

Definition (Joint p.m.f)

Let (X, Y) be a discrete bivariate random variable. Then p (x, y) is called the joint probability mass function of (X, Y) if the following conditions are satisfied.

Definition (Marginal Probability Mass Function)

Given a joint probability mass function p (x, y), then p (x) = Σ_y p(x,y) is called marginal probability mass function of X. Similarly p (y) = Σx p(x,y) is called the marginal probability mass function of Y.

Example 9.14

There are 10 tickets in a bag which are numbered 1, 2, 3, ...10. Two tickets are drawn at random one after the other with replacement.

Here, random variable X denotes the number on the first ticket and random variable Y denotes the number on the second ticket.

Joint and marginal probability density functions

As we defined in section 9.5.1 the joint probability mass function, we define the joint probability density function.

Definition:

 Let (X, Y) be a bivariate continuous random variables. The function f (x, y) is called a bivariate probability density if the following conditions are satisfied.

The marginal probability density function of X is given by

and the marginal probability density function of Y is given by

Example 9.15

Prove that the bivariate function given by f(x, y) = 

Proof:

If f is a probability density function

Therefore, f (x, y) is a probability density function.

Example 9.16

Joint p.d.f. of X,Y is f(x, y) =  then find the marginal density function of X and Y.

Solution:

Example 9.17

Joint p.d.f. of X,Y is f(x, y) =  Find the marginal density function of X and Y

Solution: