18.44 Lecture Notes - Lecture 20: Random Variable, Districts Of British India
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Last time we found that if x is geometric with rate 1 and n 0 then e[x ] = x e dx = n! This expectation e[x ] is actually well de ned whenever n > 1. The following quantity is well de ned for any > 0: ( ) := e[x ] = So ( ) extends the function ( 1)! (as de ned for strictly positive integers ) to the positive reals. Vexing notational issue: why de ne so that ( ) = ( 1)! instead of ( ) = ! At least it"s kind of convenient that is de ned on (0, ) instead of ( 1, ). The sum x of n independent geometric random variables of parameter p is negative binomial with parameter (n, p). Recall that we can approximate a poisson process of rate by tossing n coins per time unit and taking p = /n.
