Hi RonL
Thank you. Am I on the right track with this below then for normal dist MGF:
I am checking the pages for MGF and normal distribution for this formula below:
But I did not see it yet, where does this come from please?
In first point, am I doing power series expansion wrong? Because I do not know where will come from for denominator in final equation or how to see that odd moments are 0.
Thank you again
The MGF you quoted in your original question is for X ~ N(0, ): . So I don't know where you got those x's from in your power series ....??
By definition, the (2k)th moment is given by . That is, you have to differentiate the MGF 2k times and then substitute t = 0.
The power series is
Note that the kth term of this power series is .
If you think about it, the answer you want is got by differentiating this term 2k times .....
Hello Mr F
Thank you for your help, I tell you where I get my x's!:
I put in x's because I thinking like this:
t is variable so i differentiate with respect to t which I thought would give me each time so I would have:
which I equate against polynomial expression set up and differentiated at same time
I was confuse about this method because I trying to think what point I expand about which no make sense. Is it completely wrong to do this?
I was thinking that you need to be so that you get when you differentiate. Is this property of MGF that I don't understand maybe?
How does this prove that odd moments are not there?
Thank you again.
Sorry but you have the wrong idea. The MGF is function of t and so its power series expansion is a function of t. The result is given in my previous reply. The expansion is the standard expansion for an exponential function (it's expanded around the point t = 0). You might need to review this idea a bit more thoroughly.
The odd moments are zero because if you differentiate an odd number of times, all the terms in the power series expansion contain t. When you put t = 0 the result is therefore zero.
I guess the part where we say that
has expansion
I no understand that very well. I thought that MGF is proved by Power series expansion of it seems here we just take exponent and leave as it is, why is it unaffacted?
I also do not know where this assumption come from:
You know that the series expansion for e^w is .
So to get the series expansion of you can just make the substitution . Then you will get the result I stated earlier.
"I thought that MGF is proved by Power series expansion of it seems here we just take exponent and leave as it is, why is it unaffacted?"
That is done as part of the proof that .
Summary of general results:
1. The MGF of a random variable X is .
2. The nth moment of X , that is, is given by evaluated at t = 0, that is, .
In your problem, and n = 2k. It is convenient to get the (2k)th derivative of by writing it as a power series. Alternatively, you could just differentiate 2k times without using the expansion.
Do not confuse expressing the calculated MGF as a power series in order to easily do the differentiations with using a power series to prove that the nth moment of the MGF is given by the formula in 2. above.
I hope this clears things up for you.
Hi,
I'm working on what seems to be essentially the same problem. I'm stuck in one area.
I understand that the mgf can be written
but I don't understand how to get to
the -> is tripping me up.
Thanks!