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Let 13 cards be taken, at random and without replacement, from an ordinary deck of playing cards. If  is the number of spades in these 13 cards, find the pmf of . If, in addition,  is the number of hearts in these 13 cards, find the probability  What is the joint pmf of  and ?

1. Let  zero elsewhere, be the pdf of  and . Then if  , compute  and, more generally, for  What is the pdf of ? 2. Let  and  have the pdf  zero elsewhere. Find the cdf and pdf of the product 

Consider Example 2.1.3. (b) Using Part (a) and the notation in Example 2.1.3, show that   (c) Show that the following R program computes  Then use it to compute the probabilities in Part (b). Example 2.1.3

1. Let  and  have the joint pdf  zero elsewhere. Find the marginal pdfs and compute  2. Let  be two random variables with the joint pmf   By following the proof of Theorem 1.8.1, show that Theorem 1.8.1

Let the random variables  and  have the joint pmf described as follows: and  is equal to zero elsewhere. (a) Write these probabilities in a rectangular array as in Example 2.1.4, recording each marginal pdf in the “margins.” (b) What is ? Example 2.1.4

1. Let X and Y have the joint pmf  (1, 2),(2, 2), zero elsewhere. Find the correlation coefficient ρ 2. Let  and  have the joint pmf described by the following table:

Let show that And yield the means, the variances, and the covariance of the two random variables. Use this result to find the means, the variances, and the covariance of X and Y of Example 2.5.6. Example 2.5.6

Let  have joint pdf  zero elsewhere. (a) Find the marginal probability density functions of X, Y, and Z.

Let  

1. Derive the second equality in expression (2.8.7). 2. let  , zero elsewhere. Find the mean and variance of the sum Y of these four random variables.