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and next the generating function for negative binomials and i also tell you why the negative binomial distribution is actually called negative binomial to find the generating function for negative binomial iamp;#39;m going to borrow a fact from analysis the definition and the fact so the fact actually starts with the definition fix any real number and a positive integer m iamp;#39;m going to define the binomial coefficient also choose m for this scenario okay and hereamp;#39;s how iamp;#39;m going to do this so what is n choose m n choose m is n factorial over m factorial n minus m factorial if everybody is a positive integer and actually i can cancel out a lot of products from n factorial and n minus m factorial what is left is n n minus 1 down to n minus m plus 1 over m factorial because everything else cancels out if i just expand the 2 factorials n factorial and m minus m factorial and this way of writing it makes perfect sense if i replace n by a real so the definition of my b
