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 \(\sigma\):
Some of these results will be proved later.