# Thread: MGF of sample mean of IID poisson RVs

1. ## MGF of sample mean of IID poisson RVs

I am given the MGF of IID RVs as and am supposed to find the MGF of their sample mean of n such RVs.
I have am arriving at

This doesn't appear to be a valid MGF and there is another part to the question where it asks for the

I am confused where might i be making a mistake and if it is proper to have the MGF running to infinity.
Thank You

2. ## Re: MGF of sample mean of IID poisson RVs

What is the equation for MGF of sample mean of n RVs? Is it [Mx(t/n)]^n ?

Because if I use the above, then I am getting My attempt

3. ## The Following User Says Thank You to Buckeye For This Useful Post:

nsus (02-14-2017)

4. ## Re: MGF of sample mean of IID poisson RVs

Originally Posted by nsus
I am given the MGF of IID RVs as and am supposed to find the MGF of their sample mean of n such RVs.
I have am arriving at

This doesn't appear to be a valid MGF and there is another part to the question where it asks for the

I am confused where might i be making a mistake and if it is proper to have the MGF running to infinity.
Thank You
Well, Buckeye's close, however, you should end up with:

M_n(t) = ( Exp [ (Lambda/n) * ((Exp[t] - 1)) ] )^n.

Taking the limit as n->Infinity will yield the MGF you provided above in your first sentence. I hope this helps.

5. ## Re: MGF of sample mean of IID poisson RVs

I have arrived at a result similar to Buckeye and Dragan's result eludes me.
I used , where Y is the sample mean.
But i still can't figure out how should an mgf look as n tends to infinity.
Any pointers will be appreciated.

6. ## Re: MGF of sample mean of IID poisson RVs

Originally Posted by nsus

I have arrived at a result similar to Buckeye and Dragan's result eludes me.
I used , where Y is the sample mean.
But i still can't figure out how should an mgf look as n tends to infinity.
Any pointers will be appreciated.
Well, if you take the limit of the functions that either you or Buckeye provided as n-> Infinity, then the answer is Zero. Thus, if that is a satisfactory answer to you, then that is just fine with me.

7. ## The Following User Says Thank You to Dragan For This Useful Post:

nsus (02-14-2017)

8. ## Re: MGF of sample mean of IID poisson RVs

Originally Posted by Dragan
Well, if you take the limit of the functions that either you or Buckeye provided as n-> Infinity, then the answer is Zero. Thus, if that is a satisfactory answer to you, then that is just fine with me.
Thank you so much for your time.
I did a bit of reading and figured that an MGF of One would imply the RV to be equal to zero. I used the Taylor's series to solve for the limit and found that

as .
This seems to be an acceptable solution.
Thanks for your time once again.

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