# Thread: Expected value of random variables

1. ## Expected value of random variables

Let N,X1, X2... be random variables where N has a poisson distribution with mean 3 while X1, X2...each has a Poisson distribution with mean 7.
a) Determine $E[N\sum^{N}_{i=1} X_i]$
b) Determine the variance of $[\sum^{N}_{i=1} X_i]$

I have no idea where to start. Please give me a hand. Thanks

2. Hello,
Originally Posted by EitanG
Let N,X1, X2... be random variables where N has a poisson distribution with mean 3 while X1, X2...each has a Poisson distribution with mean 7.
a) Determine $E[\sum^{N}_{i=1} X_i]$
b) Determine the variance of $[\sum^{N}_{i=1} X_i]$

I have no idea where to start. Please give me a hand. Thanks
If you know no formula for this, the best way is to partition the set of events :
(assuming that N and the sequence X_i are independent, and I guess there's an extra N in what you wrote... anyway, it would be the same reasoning)

$\mathbb{E}\left(\sum_{i=1}^N X_i\right)=\sum_{k=0}^\infty k \mathbb{P}\left(\sum_{i=1}^N X_i=k\right)$

And $\mathbb{P}\left(\sum_{i=1}^N X_i=k\right)=\sum_{n=1}^\infty \mathbb{P}\left(\sum_{i=1}^N X_i=k ~,~ N=n\right)$

$=\sum_{n=1}^\infty \mathbb{P}\left(\sum_{i=1}^n X_i=k ~,~ N=n\right)=\sum_{n=1}^\infty \mathbb{P}\left(\sum_{i=1}^1 X_i=k\right)\mathbb{P}(N=n)$

The sum of Poisson variables is a Poisson variable (whose parameter is the sum of the parameters)
And you have to see what happens if N is 0. This should be stated in your problem, or conventionally, the sum would be 0.

It's a bit messy, but you can get the information you need from here...

3. It looks like you have an extra N in the expectation.
This is Wald's equation, see...
Wald's equation - Wikipedia, the free encyclopedia

Originally Posted by Moo
Hello,

If you know no formula for this, the best way is to partition the set of events :
(assuming that N and the sequence X_i are independent, and I guess there's an extra N in what you wrote... anyway, it would be the same reasoning)

$\mathbb{E}\left(\sum_{i=1}^N X_i\right)=\sum_{k=0}^\infty k \mathbb{P}\left(\sum_{i=1}^N X_i=k\right)$

And $P(\sum_{i=1}^N X_i=k)=\sum_{n=1}^\infty P(\sum_{i=1}^N X_i=k ~,~ N=n)$

$=\sum_{n=1}^\infty P(\sum_{i=1}^n X_i=k ~,~ N=n)=\sum_{n=1}^\infty P(\sum_{i=1}^{n}X_i=k)P(N=n)$

The sum of Poisson variables is a Poisson variable (whose parameter is the sum of the parameters)
And you have to see what happens if N is 0. This should be stated in your problem, or conventionally, the sum would be 0.

It's a bit messy, but you can get the information you need from here...

5. No guys that N is supposed to be in the first line. The quote ignored it for some reason