Log Normal Distribution

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##### Log Normal Distribution

    #include <boost/math/distributions/lognormal.hpp>

    namespace boost{ namespace math{

    template <class RealType = double,
              class Policy   = policies::policy<> >
    class lognormal_distribution;

    typedef lognormal_distribution<> lognormal;

    template <class RealType, class Policy>
    class lognormal_distribution
    {
    public:
       typedef RealType value_type;
       typedef Policy   policy_type;
       // Construct:
       lognormal_distribution(RealType location = 0, RealType scale = 1);
       // Accessors:
       RealType location()const;
       RealType scale()const;
    };

    }} // namespaces

The lognormal distribution is the distribution that arises when the logarithm of the random variable is normally distributed. A lognormal distribution results when the variable is the product of a large number of independent, identically-distributed variables.

For location and scale parameters m and s it is defined by the probability density function:

The location and scale parameters are equivalent to the mean and standard deviation of the logarithm of the random variable.

The following graph illustrates the effect of the location parameter on the PDF, note that the range of the random variable remains [0,+∞] irrespective of the value of the location parameter:

The next graph illustrates the effect of the scale parameter on the PDF:

##### Member Functions

    lognormal_distribution(RealType location = 0, RealType scale = 1);

Constructs a lognormal distribution with location location and scale scale.

The location parameter is the same as the mean of the logarithm of the random variate.

The scale parameter is the same as the standard deviation of the logarithm of the random variate.

Requires that the scale parameter is greater than zero, otherwise calls domain_error.

    RealType location()const;

Returns the location parameter of this distribution.

    RealType scale()const;

Returns the scale parameter of this distribution.

##### Non-member Accessors

All the usual non-member accessor functions that are generic to all distributions are supported: Cumulative Distribution Function, Probability Density Function, Quantile, Hazard Function, Cumulative Hazard Function, mean, median, mode, variance, standard deviation, skewness, kurtosis, kurtosis_excess, range and support.

The domain of the random variable is [0,+∞].

##### Accuracy

The lognormal distribution is implemented in terms of the standard library log and exp functions, plus the error function, and as such should have very low error rates.

##### Implementation

In the following table m is the location parameter of the distribution, s is it's scale parameter, x is the random variate, p is the probability and q = 1-p.

Function | Implementation Notes
--- | ---
pdf | Using the relation: pdf = e-(ln(x) - m)2 / 2s2 / (x * s * sqrt(2pi))
cdf | Using the relation: p = cdf(normal_distribtion<RealType>(m, s), log(x))
cdf complement | Using the relation: q = cdf(complement(normal_distribtion<RealType>(m, s), log(x)))
quantile | Using the relation: x = exp(quantile(normal_distribtion<RealType>(m, s), p))
quantile from the complement | Using the relation: x = exp(quantile(complement(normal_distribtion<RealType>(m, s), q)))
mean | em + s2 / 2
variance | (es2 - 1) * e2m + s2
mode | em + s2
skewness | sqrt(es2 - 1) * (2 + es2 )
kurtosis | e4s2 + 2e3s2 + 3e2s2 - 3
kurtosis excess | e4s2 + 2e3s2 + 3e2s2 - 6