The Modified Lognormal Power-Law (MLP) function is a three parameter function that can be used to model data that have characteristics of a lognormal distribution and a power-law behavior. It has been used to model the functional form of the Initial Mass Function (IMF). Unlike the other functional forms of the IMF, the MLP is a single function with no joining conditions.
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Functional form of the MLP distribution
If the random variable W is distributed normally, i.e. W ~ N (μ,σ2), then the random variable M = eW will be distributed lognormally:
The parameters
where
Mathematical Properties of the MLP distribution
Following are the few mathematical properties of the MLP distribution:
Cumulative Distribution
The MLP cumulative distribution function (
We can see that as
Mean, Variance, Raw Moments
The expectation value of
This exists if and only if α >
which is the
Var(
Mode
The solution to the equation
where u = (1/√2(ασ0-ln
Numerical methods are required to solve this transcendental equation. However, noting that if
Random Variate
The lognormal random variate is:
where
where R(0,1) is the uniform random variate in the interval [0,1]. Using these two, we can derive the random variate for the MLP distribution to be: