Compound poisson

[Shinra]

Hello,

Let a r.v N following a compound Poisson.
The parameter of the Poisson distribution [MATH]\lambda[/MATH] is given by the r.v [MATH]\gamma[/MATH] which is a continous uniform distribution on the interval [0;a] with a>0.
We recall that the density function f is given by:
[MATH] f( \lambda ) = \frac{1}{a} \mbox{if} \ 0 \leq \lambda \leq a \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0 \ \mbox{elsewhere} [/MATH]

1) Compute E(N) and Var(N) with parameter a.
2) Give the distribution of the r.v N with the individual probabilities, and with the moment generating function.

1) Can you explain me the method to compute the expected value of N please?
I don't understand well the statement but if N is a Poisson distribution, E(N)= [MATH]\lambda[/MATH] and there's no parameter a. So maybe I have to use the density but I don't know how to link them.

Thank you

 

[Romsek]

lmgtfy

 

[Shinra]

Thank you for your answer Romsek but why the question precise "with parameter a" if it's just [MATH]\lambda[/MATH] ?
By the way, is it the same to follow a compound Poisson and a poisson distribution for the mean and variance ?

 

[Romsek]

Oh I see I didn't read this carefully enough.

There are two levels of randomness going on here.

First is that [MATH]N[/MATH] is Poisson distributed with parameter [MATH]\lambda[/MATH]
The link I posted shows you how to show that

[MATH]E[N|\lambda] = Var[N|\lambda]= \lambda[/MATH]
Then we have the fact that [MATH]\lambda[/MATH] is itself a random variable that is uniformly distributed on [MATH][0,a][/MATH]
Using conditional expectation it shouldn't be too hard to show that

[MATH]E[N]=Var[N] = \dfrac a 2[/MATH]

 

[Shinra]

Thank you for the result, but I wanted to know the start of the calculation because I still don't know how to link the density function with E(N) ☹

 

[Romsek]

suppose you have a rn N with a distribution distribution [MATH]\lambda[/MATH]
and further that [MATH]\lambda[/MATH] also has a distribution

[MATH]E[N] = \displaystyle \int \limits_{\lambda}~E[N|\lambda] p(\lambda)~d\lambda[/MATH]
here that means

[MATH]E[N] = \displaystyle \int \limits_0^a \lambda \cdot \dfrac 1 a~d\lambda[/MATH]

 

[Shinra]

Thanks a lot ! I found [MATH]\frac{a}{2}[/MATH] too and [MATH]V(N)=\frac{a^2}{12}[/MATH]
2) I don't know and don't understand how to do the first part of this question.
For the moment generating function:
[MATH]M_N(t)= \int_{0}^{a} e^{tn}f(\lambda) dn = \frac{e^{ta}-1}{ta}[/MATH]
Is it good ?

 

[Shinra]

By the way, for [MATH]V(N)[/MATH], did you do [MATH]V(N)= \int_{\lambda} V[N| \lambda]f( \lambda) d\lambda[/MATH] ?
Because when I did [MATH]V(N)=E(N^2)-E(N)^2[/MATH], I didn't find the same result as you