normal random variable
For any real numbers $\mu $ and $\sigma >0$, the Gaussian probability distribution function with mean $\mu $ and variance^{} ${\sigma}^{2}$ is defined by
$$f(x)=\frac{1}{\sqrt{2\pi {\sigma}^{2}}}\mathrm{exp}\left(\frac{1}{2}{\left(\frac{x\mu}{\sigma}\right)}^{2}\right).$$ 
When $\mu =0$ and $\sigma =1$, it is usually called standard normal distribution^{}.
A random variable^{} $X$ having distribution^{} density $f$ is said to be a normally distributed random variable, denoted by $X\sim N(\mu ,{\sigma}^{2})$. It has expected value^{} $\mu $, and variance ${\sigma}^{2}$.
Cumulative distribution function
The cumulative distribution function^{} of a standard normal variable, often denoted by
$$\mathrm{\Phi}(z)=\frac{1}{\sqrt{2\pi}}{\int}_{\mathrm{\infty}}^{z}{e}^{{x}^{2}/2}\mathit{d}x,$$ 
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 bellshaped function, which is why it is often called 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 limiting distribution of averages of large numbers of samples, justified by the central limit theorem^{}.
Properties
Mean  $\mu $ 

Variance  ${\sigma}^{2}$ 
Skewness^{}  0 
Kurtosis  3 
Momentgenerating function  ${M}_{X}(t)=\mathrm{exp}\left(\mu t+{(\sigma t)}^{2}/2\right)$ 
Characteristic function^{}  ${\varphi}_{X}(t)=\mathrm{exp}\left(\mu it{(\sigma t)}^{2}/2\right)$ 

•
If $Z$ is a standard normal random variable, then $X=\sigma Z+\mu $ is distributed as $N(\mu ,{\sigma}^{2})$, and conversely.

•
The sum of any finite number of independent normal variables is itself a normal random variable.
Relations to other distributions

1.
The standard normal distribution can be considered as a Studentt distribution with infinite^{} degrees of freedom.

2.
The square of the standard normal random variable is the chisquared random variable of degree 1. Therefore, the sum of squares of $n$ independent standard normal random variables is the chisquared random variable of degree $n$.
Title  normal random variable 
Canonical name  NormalRandomVariable 
Date of creation  20130322 11:54:20 
Last modified on  20130322 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 