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Unformatted text preview: Math331, Spring 2008 Instructor: David Anderson Section 11.2 Homework Answers Homework: pg. 474, #s 2, 6, 8. 2. From number 9 of Section 11.1 (which you did), the moment generating function of a geometric RV, X i , with parameter p is M X i ( t ) = pe t 1- qe t , where q = 1- p , and the above function is only valid if t <- ln( q ). Therefore, if X 1 ,X 2 ,... ,X n are all independent geometric RVs with parameter p , we may use Theorem 11.3 to find the moment generating function of X 1 + X 2 + + X n . M X 1 + + X n ( t ) = M X 1 ( t ) M X n ( t ) = pe t 1- qe t pe t 1- qe t ( n times ) = parenleftbigg pe t 1- qe t parenrightbigg n . Using Table 3 in the Appendix shows that this is the moment generating function for a negative binomial RV with parameters n and p . The uniqueness Theorem 11.2 then shows that X 1 + + X n must be a negative binomial RV....
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