Probability and Statistics II week 14

BIVARIATE EXPECTATION There are many problems in which we are interested not only in the expected value of a random variable X, but also in the expected values of random variables X and Y. Theorem 1 If X and Y are discrete random variables and is the value of their joint probability distribution at , the expected value of is Correspondingly, if X and Y are continuous random variables and is the value of their joint probability density at , the expected value of is Example 13 With reference to example 1, find the expected value of . Solution The results of Example 1 are shown in the following table, together with the marginal totals, that is, the totals of the respective rows and columns:   x y 0 1 2 0 1 0 2 0 0 Example 14 If the joint probability distribution of X and Y is given by: (a) Construct a table showing the values of the joint distribution function of the two random variables at the 12 points (0,0), (0,1), …, (3,2). (b) Find Solution ; ; ; ; ; ; ; ; ; ; . The results are shown in the following table: x y 0 1 2 3 0 0 1 2 Example 15 If the joint probability density of X and Y is given by: Find the expected value of . Solution