Chapter 3 Random Variables Appendix (Negative Binomial Expansion)

Chapter 3 Random Variables Appendix (Negative Binomial Expansion)

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1306 INTRODUCTORY STATISTICS (SEMESTER 1 2008/2009) Negative Binomial Expansion The conventional binomial expansion is given by (1 + y ) n = 1 X r =0 n r ± y r = 1 + ny + n ( n 1) 2! y 2 + ± ± ± + n ( n 1) ± ± ± ( n r + 1) r ! y r + ± ± ± for 1 < y < 1 positive integer n . Indeed, the expansion can be applied for negative integer n as well. (1 + y ) n =
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Unformatted text preview: 1 X r =0 & & n r ± y r = 1 X r =0 ( & 1) r & n + r & 1 r ± y r = 1 & ny + n ( n + 1) 2! y 2 & n ( n + 1) ( n + 2) 3! y 3 + ± ± ± + ( & 1) r n ( n + 1) ± ± ± ( n + r & 1) r ! y r + ± ± ± which is called the negative binomial expansion. Note that we are required to show the second equality in Assignment 1. 1...
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