# PMF and MGF of this problem

#### nacho

##### Active member
is part a) simply the sum of the PMFs of the poisson and exponential random variables we are given?

I can't quite make sense of this question. Where it says "identify a distribution..."
is it looking for us to say something like a gamma random variable or a geometric random variable etc?

thank you!

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#### chisigma

##### Well-known member
is part a) simply the sum of the PMFs of the poisson and exponential random variables we are given?

I can't quite make sense of this question. Where it says "identify a distribution..."
is it looking for us to say something like a gamma random variable or a geometric random variable etc?

thank you!
The statement of the problem is not clear at 100 x 100, but what I undestand is the PMF and MGF of the number N of customers arriving in a time T. N is Poisson distributed so that the PMF is...

$\displaystyle P \{ N = n \} = \frac{(\beta\ T)^{n}}{n!}\ e^{- \beta\ T}\ (1)$

... and the MGF is...

$\displaystyle E \{ e^{N\ t}\} = \sum_{n=0}^{\infty} P \{N = n\}\ e^{n\ t} = e^{\beta\ T\ (e^t-1)}\ (2)$

Kind regards

$\chi$ $\sigma$

#### nacho

##### Active member
chisigma, a million thank yous are not enough to express my gratitude for you.

someone make this man a mod, he is legendary.

i am also happy that you also thought the question wasn't clear, that gives me some confidence

Thanks again, this is enough to get me started on the rest !

#### chisigma

##### Well-known member
... someone make this man a mod, he is legendary...
Thank for Your compliments!... regarding the 'moderation' I consider myself totally unable to cover the role of moderator because I think that, in a family of people with the ideal to promote the mathematical knowledge, the figure of moderator shouldn't be necessary. For that reason I prefer to remain 'site helper' and to continue to do my best possible to MHB...

Kind regards

$\chi$ $\sigma$