Normal Distribution

A sample is merely a collection of values randomly drawn from a common distribution which is not yet known. Such a distribution is characterized by a probability density function (PDF), and often described by parameters. A normal distribution is used to represent a family of PDF's having the same general shape, and giving rise to the discipline of statistical modeling. The normal PDF is formulated by

$\displaystyle f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left[-\frac{(x - \mu)^2}{2 \sigma^2}\right]$

where $\pi = 3.14159\ldots$ and $\exp(u)$ is the exponential function

with the base $e = 2.71828\ldots$ of the natural logarithm. The first parameter $\mu$ is called mean and it determines the center of density. The second parameter $\sigma$ is called standard deviation (SD). The normal density function

is unimodal and symmetric around $\mu$ . A small value of $\sigma$ leads to a high peak with sharp drop, and a larger value of $\sigma$ leads to a flatter shape of function.

Standard normal distribution. When the parameter $(\mu,\sigma^2) = (0, 1)$ is chosen, it becomes the standard normal distribution. The corresponding density function, denoted by $\phi(x)$ , is given by

$\displaystyle \phi(x) := \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$

and Normal Distribution Table is used to obtain the proportion $\Phi(z)$ of the area under the curve up to the value $z$

. The function $\Phi(z)$ is called the standard normal cumulative distribution function.