e-Statistics > 4480-5480 Probability and Statistics II

## Normal Distribution

From the probabilistic point of view a data set of a certain population 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) which is often described by means of parameters. The parameters are used to represent a family of functions having the same general shape, and the introduction of such parameters gave rise to the discipline of statistical modeling.

A normal density function

is determined by two parameters and . The first parameter is called mean and it determines the center of the density. The second parameter is called standard deviation (SD). The normal density function is unimodal and symmetric around . A small value of leads to a high peak with sharp drop, and a larger value of leads to a flatter shape of function.

Assuming the normal density function for a certain population, we can predict the chance that the observation from the population is between and . Having formed the density function , we can obtain such a probability, denoted by , as the area under the curve over the range between and . The calculation of this probability is what is known as integration'' in light of calculus, and can be obtained numerically by

In order to compute the probability or in the form above, you must use  -Inf'' or  +Inf'' to indicate or in appropriate boxes.

Z-score. In particular, is called the standard normal distribution. Suppose that a random variable is normally distributed with . Then is called the z-score, and the distribution of becomes the standard normal distribution . Standard normal distribution table contains the area under the standard normal curve from to . This table can be used to compute the probability involving a normal random variable.