1. ## marginal moment for random sums - question about proof

Suppose that {$\displaystyle X_j$} is a sequence of intedependent identically distributed random variables and that N is a random variable taking non-negative integer values.

If $\displaystyle Y=\Sigma_{j=1}^N$, then $\displaystyle M_Y(t)=M_N(lnM_X(t))$

Proof.

$\displaystyle M_Y(t)=E[M_{Y|N}(t|N)]=E[(M_X(t)^N]=E[e^{NXt}]=E[exp({N*ln(e^{xt})]=$
$\displaystyle =E(e^{N*lnM_X(t)})=M_N(lnM_X(t))$

The beginning and the end is from the textbook, the part in the middle with $\displaystyle E[e^{NXt}]=E[exp({N*ln(e^{xt})]$ is mine.

I am not quite sure about the last transition

$\displaystyle E[exp({N*ln(e^{xt})]=E(e^{N*lnM_X(t)})=M_N(lnM_X(t))$.

I can see how $\displaystyle E[e^{N*something}]$ becomes $\displaystyle M_N(something)$. I am not sure why $\displaystyle ln(e^{xt})=lnM_X(t)$ if there is no expected value E anywhere around (since the definition of MGF is $\displaystyle M_X(t)=E(e^{Xt}$).

2. Hello !
Originally Posted by Volga
The beginning and the end is from the textbook, the part in the middle with $\displaystyle E[e^{NXt}]=E[exp({N*ln(e^{xt})]$ is mine.

(...)

I can see how $\displaystyle E[e^{N*something}]$ becomes $\displaystyle M_N(something)$. I am not sure why $\displaystyle ln(e^{xt})=lnM_X(t)$ if there is no expected value E anywhere around (since the definition of MGF is $\displaystyle M_X(t)=E(e^{Xt}$).
That's because it's $\displaystyle E[e^{NXt}]=E[\exp(N*\ln(e^{Xt}))]$ (capital X)

P.S. : HK \o///

3. It's not so obvious to me.

Instead, I would write $\displaystyle E[exp(N*ln(e^{Xt})]=M_N(ln(e^{Xt}))=M_N(Xt)$

?? I suspect there is something here about the expected value operator that I don't know...

4. Oh right sorry, I misunderstood your question.

There is a big mistake for the third '=' sign :
$\displaystyle E\left[M_X(t)^N\right]=E\left[\left(E\left[e^{Xt}\right]\right)^N\right]$, which is completely different from what you wrote.

I would like to know what you mean by "the beginning" and "the end". Do you mean that the book only provides $\displaystyle M_Y(t)=M_N(\ln M_X(t))$ ?

5. Originally Posted by Moo
Oh right sorry, I misunderstood your question.

There is a big mistake for the third '=' sign :
$\displaystyle E\left[M_X(t)^N\right]=E\left[\left(E\left[e^{Xt}\right]\right)^N\right]$, which is completely different from what you wrote.

I would like to know what you mean by "the beginning" and "the end". Do you mean that the book only provides $\displaystyle M_Y(t)=M_N(\ln M_X(t))$ ?
This is from the book:

$\displaystyle M_Y(t)=E[M_{Y|N}(t|N)]=E[(M_X(t)^N]=E(e^{N*lnM_X(t)})=M_N(lnM_X(t))$

The mistake you pointed out is mine ))) I now understand completely, from the double expected value expression you wrote. Thank you!!