Statistical Sampling

Normal distribution

One of the most well-studied and employed distributions, the normal distribution is important, among other things, for its applicability, in part due to the central limit theorem, in describing a wide variety of phenomena, as well as for its relatively straight-forward mathematical form and properties.

It is characterized by two parameters \mu and \sigma, which are equal to the mean and the standard deviation, respectively. In addition, given these values, the normal distribution can be shown to be the continuous distribution which maximizes the differential entropy. Thus in cases when little more information is available than these values, the normal distribution is generally a fairly good candidate.

Basic properties:
Notation:
X \sim \mathrm{N}(\mu, \sigma^2)
Type:
Continuous
Parameters:
\mu \in \mathbb{R}, \sigma^2 \in \mathbb{R}_{> 0}
Variables:
X
Support:
\mathbb{R}
PDF:
f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-(x - \mu)^2/2\sigma^2}
CDF:
F(x) = P(X \le x) = \frac{1}{2} \Bigg( 1 + \mathrm{erf}\bigg( \frac{x - \mu}{\sigma\sqrt{2}} \bigg) \Bigg)
Mean:
\mu
Variance:
\sigma^2