Originally Posted by

**Katina88** Hi, I'm really bad at basic math can someone please help me complete the square ><?

$\displaystyle

\int_{-\infty}^{\infty} \frac{e^{tx}}{\sigma\sqrt{2\Pi}} e^{\frac{-(x-\mu)^2}{2 \sigma{^2}}} dx

$

Then i let, $\displaystyle z = \frac{x - \mu}{\sigma}; dz = \frac{1}{\sigma}; x = z\sigma + \mu $

$\displaystyle

\int_{-\infty}^{\infty} \frac{e^{tz\sigma + \mu}}{\sqrt{2\Pi}} e^{\frac{z^2}{2}} dz

$

hmm! i really don't know what I'm doing but I'll just keep going..

$\displaystyle

\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\Pi}} e^{\frac{z^2+ 2t(z\sigma + \mu)}{2}}e^{\frac{t^2}{2}} dz

$

yeahh i really don't know what to do but the answer is meant to be:

$\displaystyle e^{t\mu + \frac{t^2\sigma^2}{2}}$

I know that eventually I'm meant to have

$\displaystyle

\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\Pi}} e^{\frac{-(x-\mu)^2}{2 \sigma{^2}}} dx = 1

$

and then have the answer outside this integral, but i don't know how to do it. Please help me