normal random variable
For any real numbers and , theGaussian probability distribution functionwith mean and variance is defined by
When and ,it is usually called standard normal distribution.
A random variable having distribution
density is said to be anormally distributed random variable, denoted by .It has expected value
,and variance .
Cumulative distribution function
The cumulative distribution function of a standard normal variable,often denoted by
cannot be calculated in closed form in terms of the elementary functions,but its values are tabulated in most statistics books and here (http://planetmath.org/TableOfProbabilitiesOfStandardNormalDistribution),and can be computed using most computer statistical packages and spreadsheets.
Uses of the Gaussian distribution
The normal distribution is probably the most frequently used distribution.Its graph looks like a bell-shaped function, which is why it is oftencalled bell distribution.
The normal distribution is important in probability theory and statistics.Empircally, many observed distributions,such as of people’s heights, test scores,experimental errors, are found to be more or less to be Gaussian.And theoretically, the normal distribution arises as a limitingdistribution of averages of large numbers of samples, justifiedby the central limit theorem.

Properties
- •
If is a standard normal random variable,then is distributed as ,and conversely.
- •
The sum of any finite number of independent normal variables is itselfa normal random variable.
Relations to other distributions
- 1.
The standard normal distribution can be considered as a Student-t distributionwith infinite
degrees of freedom.
- 2.
The square of the standard normal random variable is the chi-squared random variable of degree 1. Therefore, the sum of squares of independent standard normal random variables is the chi-squared random variable of degree .
Title | normal random variable |
Canonical name | NormalRandomVariable |
Date of creation | 2013-03-22 11:54:20 |
Last modified on | 2013-03-22 11:54:20 |
Owner | Koro (127) |
Last modified by | Koro (127) |
Numerical id | 22 |
Author | Koro (127) |
Entry type | Definition |
Classification | msc 62E15 |
Classification | msc 60E05 |
Classification | msc 05C50 |
Classification | msc 34K05 |
Synonym | normal distribution |
Synonym | standard normal distribution |
Synonym | bell distribution |
Synonym | bell curve |
Synonym | Gaussian |
Related topic | AreaUnderGaussianCurve |
Related topic | JointNormalDistribution |