Tuesday, 30 August 2016

Show that G(s)=1alpha(1s)beta is the probability generating function of a nonnegative integer valued random variable




I'm working on the following exercise:




Show that G(s)=1α(1s)β is the probability generating function of a nonnegative integer valued random variable when α,β(0,1).




I tried the following:



The probability generating function of a discrete random variable X is defined by GX(s)=E(sX)=k=0skP(X=k), and thus G(k)X(0)=k!P(X=k), where G(k)X(s) denotes the k'th derivative with respect to s. Thus P(X=k)=G(k)X(0)k!. Working this out I find:




P(X=0)=G(0)X(0)0!=G(0)X(0)=GX(0)=1αP(X=1)=G(1)X(0)1!=αβP(X=2)=G(2)X(0)2!=αβ(1β)2!  P(X=n)=G(n)X(0)n!=αβ(1β)(2β)(nβ)n!



If α,β(0,1) it follows that all probabilities P(X=k), for kN{0} are in (0,1). For this nonnegative integer valued random variable I will assume that P(X=x)=0 for xN{0}.




To show that this is indeed the probability generating function of some nonnegative integer valued random variable I have to show that k=0P(X=k)=1α+k=1αβ(1β)(2β)(kβ)k!=1α+αk=1β(1β)(2β)(kβ)k! equals 1. I didn't succeed to show this, but I have the feeling I have to use the Binomial Theorem somehow. Any ideas? Thanks in advance!


Answer



For α,β(0,1), your G satisfies




  1. 0<G(s)<1 for all s[0,1).

  2. G is infinitely differentiable on [0,1) with G(n)0.

  3. lim.




So G is a probability generating function.






Edit: You are trying to use that G(s)=\sum_{n=0}^{\infty}\frac{G^{(n)}(0)}{n!}s^n, but you do not have to calculate the power series on the RHS and plug in s=1. You already have its closed form (the LHS) and you can do this directly.


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