# Normal Distribution

Pocket

Regarding Statistical Distributions, there are many different types. Among them, the most important one is the Normal (Gaussian) Distribution. The corresponding PDF (Probabilistic Density Function) is given as follows:

$f(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp \Bigl( -\frac{(x-\mu)^2}{2\sigma^2}\Bigr)$

Here, $\mu$ indicates Average (mean), and $\sigma$ denotes Standard Deviation.

The below Figure shows the Probabilistic Density Function of Normal Distribution, in case that Average is $\mu=170$ and Standard Deviation is $\sigma=6$. This example may correspond to the body male height for one population.

As you can see from Figure, the peak of the distribution is around $\mu=170$㎝. Roughly speaking, the width of the hill at the half-height corresponds to the Standard Deviation.

As a check, we can compute the expectation E and Variance V, which give the following results:

$E=\int_{-\infty}^{\infty}x p(x) dx=\mu$

and

$V=\int_{-\infty}^{\infty}(x-\mu)^2 p(x)dx=\sigma^2$.