# Any idea on how to integrate the following?

1. May 28, 2014

### Amannequin

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

The c.d.f. of a random variable X is given by the following formula

F(x) = 1 - e-x2 x≥0

Find M(t). Moment generating function.

3. The attempt at a solution

I've found the pdf as

f(x) = F'(x) = 2xe-x2 x≥0

To determine M(t) I need to do the following

M(t) = E(etX) = ∫etx 2xe-x2

Then I'm not sure how to go about integrating. Any ideas or clues to point me in the right direction will be greatly appreciated. Thanks.

2. May 28, 2014

### jbunniii

Try completing the square: $x^2 - tx = x^2 - tx + (t/2)^2 - (t/2)^2 = (x - t/2)^2 - (t/2)^2$, so
$$\int e^{tx}2xe^{-x^2}dx = e^{(t/2)^2} \int 2x e^{-(x-t/2)^2} dx$$
which should be manageable with a bit more work.