Integration of the normal density function

From Wikipedia, the free encyclopedia

Main article: Normal distribution

The probability density function for the normal distribution is given by

f(x;\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}} \, \exp \left( -\frac{(x- \mu)^2}{2\sigma^2} \right),

where μ is the mean and σ the standard deviation.

By the definition of a probability density function, f must integrate to 1. That is,

I = \int_{-\infty}^{\infty} f(x)\, dx = 1.

However, this integration is not straight-forward, since f does not have an antiderivative in closed form. For the special case when μ = 0 and σ = 1, one method is to pass to the related double integral

\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{1}{2\pi} \exp \left( \frac{-x^2-y^2}{2} \right) \, dx \, dy = I^2.

This double integral in cartesian coordinates can be converted to the following integral in polar coordinates

\int_0^{2\pi} \int_0^{\infty} \frac{r}{2\pi} \exp (-r^2/2) \, dr \, d\theta = \int_0^{\infty} r \exp (-r^2/2) \, dr

which can be evaluated using the substitution u = − r2 / 2 to yield 1, the desired result.