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May 7th, 2016, 08:15 AM   #1
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From: london

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Moment generating function

If I have a sequence of independent random variables having a standard normal distribution. Suppose N has a poisson distribution with mean $\displaystyle \lambda$ and is independent of this sequence.

Define a new random variable W = sum of Xi from i=0 to N.

Show moment generating function of W is:

$\displaystyle Mw(\phi) = e^{\lambda(e^{0.5\phi^2 -1)}} $

I thought to get this I would just take the moment generating function of a poission i.e $\displaystyle e^{\lambda(e^{\phi}-1})$ to the power of N, but its not the answer?
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May 22nd, 2016, 01:52 AM   #2
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Hey calypso.

I'd take a look at the MGF identity for adding two Normal distributions conditional on the value of N.

Once you get the probability density function you can take the MGF of that.

It means you have to figure out a conditional probability given the value of N (so you look at the probability of getting a particular value of N) and then use that value to determine your MGF.

If you are looking at overall expectations then you can use the total expectation laws which look at finding expectations of conditional random variables.

Please post your attempts so we can have a look at them and give further guidance.
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