Normal distributions
The family of normal distributions is another family of distributions whose shapes are flexible enough to be used as models for many practical variables. Normal distributions were introduced earlier but we will now define the Normal distribution more formally.
Definition
A random variable, \(X\), is said to have a normal distribution,
\[ X \;\; \sim \; \; \NormalDistn(\mu,\; \sigma^2) \]if its probability density function is
\[ f(x) \;\;=\;\; \frac 1{\sqrt{2\pi}\;\sigma} e^{- \frac{\large (x-\mu)^2}{\large 2 \sigma^2}} \qquad \text{for } -\infty \lt x \lt \infty \]The diagram below illustrates the range of possible shapes of Normal distributions.
Shape of the Normal distribution
Standard normal distribution
One particular member of the family of normal distributions has a particularly simple pdf.
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 is therefore
\[ f(z) \;\;=\;\; \frac 1{\sqrt{2\pi}} e^{- \frac{\large z^2}{\large 2}} \qquad \text{for } -\infty \lt x \lt \infty \]