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The Lognormal Distribution
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The Lognormal Distribution
The lognormal distribution is used to model variables whose
logarithm has a normal
distribution. More generally, the lognormal distribution can be
applied to variables that are positive and have some very large
values. Examples of variables that can be modeled with a lognormal
distribution include:
- the concentration of air pollutants;
- the down time of faulty equipment;
- the concentration of minerals in deposits.
The lognormal distribution is closely related to the normal distribution. The logarithm of
a variable with a lognormal distribution has a normal distribution.
We will refer to this transformed distribution as the associated
normal distribution in the rest of this section.
The lognormal distribution has a location parameter and a scale
parameter, usually denoted by the Greek letters μ and σ.
They correspond to the mean and the standard deviation of the
associated normal distribution. The probability density function
is:
The lognormal distribution is implemented by the
LognormalDistribution class. It has two constructors.
The first constructor has two parameters. The first parameter is
the location parameter, and corresponds to the mean of the
associated normal distribution. The second parameter is the scale
parameter and corresponds to the standard deviation of the
associated normal distribution.
The following constructs the lognormal distribution with
location parameter 6.8 and scale parameter 4.1:
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LognormalDistribution lognormal = new LognormalDistribution(6.8, 4.1); |
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Dim lognormal As LognormalDistribution = New LognormalDistribution(6.8, 4.1) |
The LognormalDistribution class has two specific
properties,
LocationParameter and
ScaleParameter, which return the location and scale
parameters of the distribution.
If a variable is assumed to have a lognormal distribution, then
the order and scale parameters of the distribution can be estimated
using the method of maximum likelihood or the method of matching
moments. This is done by estimating the parameters of the normal
distribution for a logarithmic transformation of the variable. 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.
LognormalDistribution 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 = LognormalDistribution.GetRandomVariate(random, 6.8, 4.1); |
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Dim random As MersenneTwister = New MersenneTwister()
Dim variate As Double = LognormalDistribution.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 Normal Distribution Previous: The Logistic Distribution Contents
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