Normal Distribution

[dttom]

This is not a homework question, that I work on integration of normal distribution and found some difficulties.

The difficulty could be summarised in integrating this:

f(x) = e^(x^2), I am not familiar to integrating exponential function to power x. Hope could get some help, thanks.

[Justonium]

I'm not sure how to intergrate that; when I try 'u' substitution, representing the (x^2) as 'u', I can't get the 'du' in there:

du = (d/dx)u = (d/dx)(x^2) = 2xdx

There is no 'x' term in the expression, however, so that is where I get stuck.

[timo]

I don't think an analytic solution for [math]\int_a^b e^{-x^2} \, dx[/math] is known. But the integral [math]\int_{-\infty}^\infty e^{-x^2} \, dx[/math] exists and is well known (except to me who never remembers such things) and various ways to calculate is should be available online. The term you want to google for is probably "gaussian integral".

Edited July 16, 2010 by timo

[Bignose]

You are looking for the "error function": http://en.wikipedia.org/wiki/Error_function