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
(n - 1)!