Expected Value about Normal Distribution

I could have completed this post in the last month, however I was too exhausted to write this article yesterday (the last day of Feb).

Recently, I am addicted in Probability & Statistics theorem and calculus. I found there is a lot of interesting formula derivation about Normal Distribution (aka Gaussian Distribution), so I’d like to proof it by myself. Here are several methods I tried:

The first formula is the expected value of lognormal distribution. We know, for a continuous function:

And the Probability Density Function (PDF) of lognormal distribution function is:

Hence, we have:

Because of,

For now, we are going to proof the above formula, that why the integral of normal distribution is 1.

We define:

And we convert the Cartesian coordinates to polar coordinates: