We are given a sequence of random varibles: $X_1,X_2,....X_n$ that each $X_i$ distribution is $Ber(p)$

We have : $N$ ~ $Poi( \lambda )$

We are also given : $S=X_1+X_2+....+X_n$ and: $V=N-S $

I need to compute this formula: $E[ e^{ t_{1}S +t_{2}V} | N=n]$

My first guess was that it correspond to MGF of $S$ and $V$ separately but I dont know how can I simplify this formula of conditional expectation when I have two variables in the left side... I cant find the trick to solve it!

Also seem to me that S is Binomial distribution with $(n,p)$ am I right?


If you're conditioning on N, then the fact that it's poisson distributed is irrelevant. Since $N=n$, V is essentially the number of "failures" while S is the number of "successes", with $V + S = N\;or\;n$. Therefore, this reduces your problem to:

$E[ e^{ t_{1}S +t_{2}V} | N=n] = E[ e^{ t_{1}S +t_{2}(n-S)}] =E[e^{(t_1-t_2)S+t_2n}]=e^{t_2n}E[e^{(t_1-t_2)S}]$ Where S is a binomial distribution with parameters n,p (as you conjectured), and the rest are just constants.

Therefore, $e^{t_2n}E[e^{(t_1-t_2)S}]=e^{t_2n}\sum\limits_{i=0}^n \left( e^{(t_1-t_2)i}Bin(i;n,p)\right)$

The fact that N is poisson only comes into play if you wanted the unconditional expected value.


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