(II) The Normal Probability Density:



8.2. The Normal Probability Density

The normal probability density, also called the Gaussian density, might be the most commonly used probability density function in statistics.

Normal Probability Density Function:

A random variable X taking values in [pic] has the normal probability density function [pic] if

[pic],

where

[pic]

The graph of [pic] is

[pic]

Properties of Normal Density Function:

a)

[pic]

and

[pic]

b)

[pic]

c) X is a random variable with the normal density function. X is denoted by

[pic]

d) The standard deviation determine the width of the curve. The normal density with larger standard deviation would be more dispersed than the one with smaller standard deviation. In the following graph, two normal density functions have the same means but different standard deviations, one is 1 (the solid line) and the other is 2 (the dotted line):

[pic]

e) The normal density is symmetric with respect to mean. That is,

[pic].

Also,

[pic].

f) The probability of a normal random variable follows the empirical rule introduce previously. That is,

[pic]

i.e., the probability of X taking values within one standard deviation is about 0.68, within two standard deviations about 0.95, and within three standard deviation about 1.

Standard Normal Probability Density Function:

A random variable Z, taking values in [pic] has the standard normal probability density function [pic] if

[pic],

where

[pic].

Note: we denote Z as [pic]

The probability of Z taking values in some interval can be found by the normal table. The probability of Z taking values in [0,z], [pic]can be obtained by the normal table. That is,

[pic]

The graph is given below:

[pic]

Example:

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Example:

[pic]. What is x?

[solutions:]

[pic]

[pic]

[pic]

[pic]

Computing Probabilities for any Normal Random Variable:

Once the probability of the standard normal random variable can be obtained, the probability of any normal random variable (not standard) can be found via the following important rescaling:

[pic]

Example:

Let [pic]. Please find [pic].

[solutions:]

[pic] Then,

[pic]

Online Exercise:

Exercise 8.2.1

Exercise 8.2.2

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