# Moment generating function

1. May 5, 2010

### donutmax

Is the following correct?

$$M(t)=1+t\mu'_1+\frac{t^2}{2!}\mu'_2+\frac{t^3}{3!}\mu'_3+... =\sum_{n=0}^{\infty} \frac {E(Y^n)t^n}{n!}$$

where
$$\mu'_n=E(Y^n)$$

Last edited: May 5, 2010
2. May 5, 2010

### thrillhouse86

I think that will only be true if your moment generating function is an exponential - in which case you are just doing a Taylor expansion. This is true for a Wiener process - but I don't think the relation you have holds in general.

Regards,
Thrillhouse86

3. May 5, 2010

$$m_Y(t) = E(e^{ty}) = \int e^{ty} \, dF(y) = \int \sum_{n=0}^\infty \frac{(ty)^n}{n!}\,dF(y) = \sum_{n=0}^\infty \left(\int \frac{(ty)^n}{n!} \, dF(y) \right)$$