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The Normal Distribution
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The Normal Distribution
The normal distribution is the most well-known and most widely
used statistical distribution and can be applied to a wide range of
phenomena. Some statistical tests require that the data follow a
normal distribution.
The normal distribution has two parameters usually denoted by
the Greek letters μ and σ. μ is the mean of the
distribution and acts as the location parameter. σ is the
standard deviation of the distribution and acts as the scale
parameter. The probability density function is:
The normal distribution is also known as the Gaussian
distribution.
The normal distribution is implemented by the NormalDistribution
class. It has two constructors. The first constructor takes two
parameters. The first parameter is the mean, which acts as the
location parameter. The second parameter is the standard deviation
and acts as the scale parameter.
The following constructs the normal distribution with mean 6.8
and standard deviation 4.1:
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NormalDistribution normal = new NormalDistribution(6.8, 4.1); |
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Dim normal As NormalDistribution = New NormalDistribution(6.8, 4.1) |
If a variable is assumed to have a normal distribution, then the
parameters of the distribution can be estimated using the method of
maximum likelihood or the method of matching moments. Both methods
produce the same results. The second constructor performs this
calculation. It takes one parameter: a NumericalVariable
whose distribution is to be estimated.
Note that parameter estimation says nothing about how well the
estimated distribution fits the variable's distribution. Use one of
the goodness-of-fit tests to verify the appropriateness of the
choice of distribution.
NormalDistribution class has one specific property:
the Standard
property returns a NormalDistribution object that
represents the standard normal distribution with mean 0 and
standard deviation 1. The location and scale parameters are given
by the Mean
and
StandardDeviation properties.
NormalDistribution has one static (Shared
in Visual Basic) method,
GetRandomVariate, which generates a random variate
using a user-supplied uniform random number generator. The second
and third parameters are the location and scale parameters of the
distribution.
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MersenneTwister random = new MersenneTwister();
double variate = NormalDistribution.GetRandomVariate(random, 6.8, 4.1); |
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Dim random As MersenneTwister = New MersenneTwister()
Dim variate As Double = NormalDistribution.GetRandomVariate(random, 6.8, 4.1) |
The above example uses the Mersenne
Twister to generate uniform random numbers.
For details of the properties and methods common to all
continuous distribution classes, see the topic on ContinuousDistribution
class.
Up: Continuous Probability Distributions Next: The Pareto Distribution Previous: The Lognormal Distribution Contents
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