normal random variable

For any real numbers μ and σ>0, the Gaussian probability distribution function with mean μ and varianceMathworldPlanetmath σ2 is defined by


When μ=0 and σ=1, it is usually called standard normal distributionMathworldPlanetmath.

A random variableMathworldPlanetmath X having distributionPlanetmathPlanetmathPlanetmath density f is said to be a normally distributed random variable, denoted by XN(μ,σ2). It has expected valueMathworldPlanetmath μ, and variance σ2.

Cumulative distribution function

The cumulative distribution functionMathworldPlanetmath 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 statisticsMathworldMathworldPlanetmath books and here (, 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 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 theoremMathworldPlanetmath.

Figure 1: Graph of densities of the normal distribution for various values of the standard deviationMathworldPlanetmath σ


Mean μ
Variance σ2
SkewnessMathworldPlanetmath 0
Kurtosis 3
Moment-generating function MX(t)=exp(μt+(σt)2/2)
Characteristic functionMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ϕX(t)=exp(μit-(σt)2/2)
  • If Z is a standard normal random variable, then X=σZ+μ is distributed as N(μ,σ2), and conversely.

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

Relations to other distributions

  1. 1.

    The standard normal distribution can be considered as a Student-t distribution with infiniteMathworldPlanetmath degrees of freedom.

  2. 2.

    The square of the standard normal random variable is the chi-squared random variable of degree 1. Therefore, the sum of squares of n independent standard normal random variables is the chi-squared random variable of degree n.

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