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Negative Binomial Lecture

Negative Binomial Lecture - LectureSep24 Background:: n...

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Lecture Sep 24 Review Geometric Distribution. Background: Generalizing binomial coefficients: First note that for any positive integer , n we can write ( ) ( ) ( ) 1 1 ! . ! ! ! n n n n k n k k n k k + = = Therefore, for any real number , α it seems reasonable to define the “binomial coefficient” ( ) ( ) 1 1 . ! k k k α α α α + = Calculus Binomial Theorem. Let α be a nonzero real number. For all real numbers x such that 1, x < we have ( ) 0 1 . k k x x k α α = + = (Proof uses Taylor’s Theorem with remainder) Now, onto the negative binomial distribution, which we cover in better detail than our text. Model: A coin comes up heads with probability . p We toss the coin repeatedly until we get r heads. We let X be the toss number of the r th head and Y be the number of tails preceding the r th head. Then according to what book you are reading either X or Y has the negative binomial distribution.

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