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单词 NormalRandomVariable
释义

normal random variable


For any real numbers μ and σ>0, theGaussian probability distribution functionwith mean μ and varianceMathworldPlanetmath σ2 is defined by

f(x)=12πσ2exp(-12(x-μσ)2).

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 anormally 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

Φ(z)=12π-ze-x2/2𝑑x,

cannot be calculated in closed form in terms of the elementary functions,but its values are tabulated in most statisticsMathworldMathworldPlanetmath 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 theoremMathworldPlanetmath.

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

Properties

  • 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 itselfa normal random variable.

Relations to other distributions

  1. 1.

    The standard normal distribution can be considered as a Student-t distributionwith 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.

Titlenormal random variable
Canonical nameNormalRandomVariable
Date of creation2013-03-22 11:54:20
Last modified on2013-03-22 11:54:20
OwnerKoro (127)
Last modified byKoro (127)
Numerical id22
AuthorKoro (127)
Entry typeDefinition
Classificationmsc 62E15
Classificationmsc 60E05
Classificationmsc 05C50
Classificationmsc 34K05
Synonymnormal distribution
Synonymstandard normal distribution
Synonymbell distribution
Synonymbell curve
SynonymGaussian
Related topicAreaUnderGaussianCurve
Related topicJointNormalDistribution
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更新时间:2025/5/4 13:03:55