# Basics of moment generating functions

1. Apr 9, 2013

### Mentallic

I missed my class that introduced us to moment generating functions, and my notes are missing some pretty essential parts to helping me understand them, so here it is:

1. The problem statement, all variables and given/known data

Find the moment generating function of $f_X(x)=e^{-x}, x>0$

2. Relevant equations

$$E(X^R)=\int_{-\infty}^{\infty}X^Rf_X(x)dx$$

R must be a natural number I believe.

$$m_X(u) = E(e^{uX})$$

3. The attempt at a solution

I'm unsure if the expectation is applied in the same way, in other words, would this be correct?

$$m_X(u) = E(e^{ux})$$

$$= \int_{-\infty}^{\infty}e^{uX}f_X(x)dx$$

$$= \int_{0}^{\infty}e^{ux}e^{-x}dx$$

$$= \int_{0}^{\infty}e^{x(u-1)}dx$$

And I'm not quite sure what the u in this case is either.

2. Apr 9, 2013

### vela

Staff Emeritus
3. Apr 9, 2013

### Mentallic

Oh awesome, thanks vela.

I can also see that the expectation can be found quite easily after deriving the moment generating function, which is good because I'll be needing it