Can PMF and MGF Be Directly Summed for Poisson and Exponential Variables?

[nacho-man]

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!

 

[chisigma]

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-man]

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]

... 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

$\chi$ $\sigma$

 

[blue_raver22]

I can provide a response to the content of this question. The PMF (probability mass function) and MGF (moment generating function) are both mathematical tools used in probability and statistics to describe the distribution of a random variable. The PMF is used for discrete random variables, while the MGF is used for continuous random variables.

In the context of this problem, it appears that we are given two random variables, a Poisson and an exponential, and asked to find the PMF and MGF for this problem. The PMF of a Poisson random variable is given by the formula P(X=k) = (e^-λ * λ^k) / k!, where λ is the parameter of the Poisson distribution. The PMF of an exponential random variable is given by the formula f(x) = λe^-λx, where λ is the parameter of the exponential distribution.

To find the PMF and MGF for this problem, we would need to use the appropriate formulas for each of the given random variables and combine them in some way. However, without more information or context, it is difficult to determine the exact method or distribution that would be used to find the PMF and MGF for this problem.

Regarding the second question, it is not clear what is being asked. It is possible that the question is asking for us to identify a distribution that could be used to model the given problem, such as a gamma or geometric distribution. However, without more information about the problem, it is not possible to accurately identify a specific distribution.