HOMEWORK SOLUTIONS FOR MATH 5651 HOMEWORK FOR WEEK 13 OF FALL 2011 Summary: HOMEWORK SOLUTIONS FOR MATH 5651 HOMEWORK FOR WEEK 13 OF FALL 2011 Text: de Groot and Schervish, 4th edition Assignment: §5.7:3,6,7,9,15; §5.10: 3,4,5,7; §6.2:5,6,8, §6.3: 2,5,6 Problem 3 of §5.7 I omit these sketches. These are easy to see on a graphing calculator, for example. Problem 6 of §5.7 We are given that X1, . . . , Xn form a random sample of size n from the ex- ponential distribution of parameter . (In the problem as stated in the text the parameter is called but I prefer to change the letter since otherwise gets overused in this problem.) We are asked to calculate the distribution of the sample mean Xn = (X1 + · · · + Xn)/n. Put Sn = X1 + · · · + Xn. Let fn(s) be the p.d.f. of Sn. Since Sn is the sum of n independent exponentially distributed random variables of parameter , we know that the distribution of Sn is a gamma distribution with parameters = n and = , by Theorem 5.7.7 of the text. Plugging into the formula for a gamma distribution we have fn(s) = n (n - 1)! sn-1 Collections: Mathematics