The family of normal distributions is flexible enough to be used as a model for many practical variables.

Definition

A random variable, \(X\), is said to have a normal distribution,

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

if its probability density function is

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

Normal distributions are symmetric and the two parameters only affect the centre and spread of the distribution.

Standard normal distribution

Definition

A standard normal distribution is one whose parameters are \(\mu = 0\) and \(\sigma = 1\),

\[ Z \;\; \sim \; \; \NormalDistn(0,\; 1) \]

A random variable, \(Z\) with a standard normal distribution is often called a z-score.

If \(Z\) has a standard normal distribution, its pdf has a particularly simple form:

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