(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the MGF (Moment generating function) of the

a. geometric distribution

b. negative binomial distribution

2. Relevant equations

geometric distribution: [tex] f(x)=p^x(1-p)^{x-1} [/tex] where x=1,2,3...

negative binomial distribution: [tex] f(x)= \frac{(x-1)!}{(x-r)!(r-1)!}p^r(1-p)^{x-r} [/tex] where x=r, r+1, r+2...

MGF= [tex] E(e^{tx}) [/tex]

3. The attempt at a solution

a. [tex] \sum_{x=1}^{\infty}e^{tx}p^x(1-p)^{x-1} [/tex]

let [itex] q=1-p [/itex]

[tex] \sum_{x=1}^{\infty}e^{tx}p^xq^{x-1} [/tex]

[tex] \sum_{x=0}^{\infty}(pe^t)q^x [/tex]

[tex] =\frac{pe^t}{1-q} [/tex]

that's as close as I can get to approximating the solution,

but the book says the answer is [tex] \frac{pe^t}{1-qe^t} [/tex]

b. [tex] \sum_{x=r}^{\infty}\frac{(x-1)!}{(x-r)!(r-1)!}e^{tx}p^rq^{x-r} [/tex] where q=1-p

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# Find the MGF of geometric,neg binomial dist.

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