The Normal Distribution and Density Functions

A fundamental role in probability theory is played by the functions and , defined as follows: for any real number

\begin{align} \phi(x) & =\frac{1}{\sqrt{2 \pi}} e^{-\frac{x^{2}}{2}} \tag{6.1} \\ \Phi(x) & =\int_{-\infty}^{x} \phi(y) d y=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{x} e^{-\frac{y^{2}}{2}} dy. \tag{6.2} \end{align} 

Because of their close relation to normal probability laws is called the normal density function and , the normal distribution function . These functions are graphed in Figs. 6A and 6B , respectively. The graph of is a symmetric bell-shaped curve. The graph of is an -shaped curve. It suffices to know these functions for positive , in order to know them for all , in view of the relations (see theoretical exercise 6.3 )

so that is a probability density function.

The importance of the function arises from the fact that probabilities concerning random phenomena obeying a normal probability law with parameters and are easily computed, since they may be expressed in terms of the tabulated function . More precisely, consider a random

phenomenon whose probability law is specified by the probability density function , given by (4.11) . The corresponding distribution function is given by

\begin{align} F(x) & =\frac{1}{\sqrt{2 \pi} \sigma} \int_{-\infty}^{x} e^{-\frac{1}{2}\left(\frac{y-m}{\sigma}\right)^{2}} d y \tag{6.6} \\ & =\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{(x-m) / \sigma} e^{-\frac{y^{2}}{2}} d y=\Phi\left(\frac{x-m}{\sigma}\right) \end{align} 

Consequently, if is an observed value of a numerical valued random phenomenon obeying a normal probability law with parameters and , then for any real numbers and (finite or infinite, in which ), 

The following example illustrates the use of (6.7) in solving problems involving random phenomena obeying normal probability laws.

Theoretical Exercises

Exercises