Normal distributions are particularly important in statistics. A particular distribution from this family is specified by the values of two parameters, usually denoted by \(\mu\) and \(\sigma\).

\[ X \;\; \sim \; \; \NormalDistn(\mu,\; \sigma^2) \]

Confusingly, the second parameter of the normal distribution is sometimes written "\(\sigma\)" instead of "\(\sigma^2\)". We will try to be explicit about whether \(\sigma\) or \(\sigma^2\) is intended, such as

\[ X \;\; \sim \; \; \NormalDistn(\mu=12,\; \sigma^2=5) \]

Shape of a normal distribution

A normal distribution's pdf has a relatively complex formula,

\[ f(x) = \frac 1 {\sqrt{2\pi}\;\sigma} e^{\large -\frac 1{2\sigma^2} (x-\mu)^2 } \quad\quad \text{for } -\infty \lt x \lt +\infty \]

The normal distribution's pdf will be treated mathematically later in the e-book.

At this point, we simply state some of its properties without proof. Its shape is determined by the values of the two parameters, \(\mu\) and \(\sigma\):

Symmetry
All normal distributions are symmetric with their maximum pdf at \(\mu\).
Mean and median
For any symmetric distribution, both the mean and median are at the point of symmetry. This holds for normal distributions so, in particular,
\[E[X] = \mu\]
Variance
The variance of the normal distribution is
\[\Var(X) = \sigma^2\]
Shape
Other than their centre and spread of values, all normal distributions have pdfs with the same basic shape. Many authors call this "bell-shaped" though the peak at \(\mu\) is sharper than most bells!

Some of these results will be proved later.