An Introduction to Stochastic Modeling, Fourth Edition by Mark A. Pinsky, Samuel Karlin

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By Mark A. Pinsky, Samuel Karlin

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X and its number is Y. Determine the conditional distribution of X given Y = y, for y = 1, 2, . . 3 Let X and Y denote the respective outcomes when two fair dice are thrown. Let U = min{X, Y}, V = max{X, Y}, and S = U + V, T = V − U. (a) Determine the conditional probability mass function for U given V = v. (b) Determine the joint mass function for S and T. 4 Suppose that X has a binomial distribution with parameters p = 12 and N, where N is also random and follows a binomial distribution with parameters q = 14 and M = 20.

53) Fixing the limits of integration leads to the gamma function, defined by ∞ xα−1 e−x dx, (α) = for α > 0. 55) and therefore, for any integers k, (k) = (k − 1)(k − 2) · · · 2 · (1). 55) shows that the gamma function at integral arguments is a generalization of the factorial function, and (k) = (k − 1)! for k = 1, 2, . . 56) provides n+ 1 2 = 1 × 3 × 5 × · · · × (2n − 1) √ π, 2n for n = 0, 1, . . 59) Stirling’s formula is the following important asymptotic evaluation of the factorial function: n!

If the sum is 4, 5, 6, 8, 9, or 10, then further rolls are required to resolve the game. , the dice are rolled repeatedly until either a sum of 4 reappears or a sum of 7 is observed. If the sum of 4 appears first, the roller wins; if the sum of 7 appears first, he or she loses. Consider repeated rolls of the pair of dice and let Zn for n = 0, 1, . . be the sum observed on the nth roll. Then, Z0 , Z1 , . . are independent identically distributed random variables. If the dice are fair, the probability mass function is pZ (2) = pZ (3) = pZ (4) = pZ (5) = pZ (6) = pZ (7) = 1 , 36 2 , 36 3 , 36 4 , 36 5 , 36 6 , 36 pZ (8) = pZ (9) = pZ (10) = pZ (11) = pZ (12) = 5 , 36 4 , 36 3 , 36 2 , 36 1 .

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