Probability and Statistics II week 13

MARGINAL DISTRIBUTIONS Definition 5 If X and Y are discrete random variables and is the value of their joint probability distribution at , the function given by: for each x within the range of X is called the marginal probability distribution function of X. Correspondingly the function given by: for each y within the range of Y is called the marginal probability distribution function of Y. Example 7 The joint probability distribution function of random variables X and Y are given in the table below: 1 2 3 1 2 3 Determine the marginal probability distribution function of: i. ii. Solution 1 2 3 1 2 3 (i) The marginal probability distribution function of is given by When When When Hence the marginal probability distribution function of X is: x 1 2 3 (ii) The marginal probability distribution function of is given by When When When Hence the marginal probability distribution function of Y is: y 1 2 3 When X and Y are continuous random variables, the probability distributions are replaced by probability densities, summations are replaced by integrals and we get