Normal distribution - Knowlpedia

For $\mu\in\mathbb R$ and $\sigma>0$ , the normal distribution $\mathcal N(\mu,\sigma^2)$ is the probability distribution on $\mathbb R$ with density

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

with respect to Lebesgue measure.

Parameters and examples

The parameter $\mu$ is the mean and $\sigma^2$ is the variance. The standard normal distribution is $\mathcal N(0,1)$ . Affine transformations satisfy $aX+b\sim\mathcal N(a\mu+b,a^2\sigma^2)$ when $X\sim\mathcal N(\mu,\sigma^2)$ and $a\ne0$ .

References

Geoffrey Grimmett and David Stirzaker, Probability and Random Processes, 3rd ed., Oxford University Press, 2001, §7.2.