# HW4 - 1-4. Casella &amp;amp; Berger 2.34, 2.36, 3.4, 3.6 5....

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Unformatted text preview: 1-4. Casella & Berger 2.34, 2.36, 3.4, 3.6 5. Show that if X has Poisson distribution with mean λ, and Y has Poisson distribution with mean µ, then X + Y also has Poisson distribution with mean λ + µ. Hint: use MGF. 6. Show that the MGF of the negative binomial distribution with parameters r and p with pmf P (X = i) = i−1 r p (1 − p)i−r , r−1 (i.e., the number of times we have to toss a coin with probability p in order to get r heads) is r et p . MX (t) = 1 − (1 − p)et Does the MGF MX (t) look like (MY (t))r of some other distribution (and if so, which one)? 7. Let X1 , . . . , Xn be independent exponentials with pdf fXi (x) = 1/β exp(−x/β ), x > 0 for some constant β > 0. Using the MGF, show that the sum Y = X1 + . . . + Xn has distribution Gamma such that fY (y ) = Γ(n1)β n xn−1 e−x/β . Hint: Read example 2.3.8 in the book for the MGF of a Gamma distribution. Then use the properties of MGFs for the sum of independent random variables to obtain the MGF of the sum of the exponentials. 8. A three-man jury has two members, each of whom independently has probability p of making the correct decision, and a third member who ﬂips a fair coin for each decision. The jury decides according to the majority vote. Consider a different jury, which has only 1 juror with probability p of making the right decision. Which jury has a better chance of making the right decision? 9. Chuck-a-Luck is a gambling game often played at carnivals and gambling houses. A player may bet on any one of the numbers 1, 2, 3, 4, 5, 6. Three dice are rolled. If the player’s number appears on one, two or three of the dice, he receives respectively one, two, or three times the money he bet, plus the original stake. What is the player’s expected loss per unit stake? 1 ...
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## This note was uploaded on 01/16/2011 for the course STAT 213 taught by Professor Ioannam during the Fall '09 term at Duke.

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