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.
If the random variable W is distributed normally, i.e. W ~ N (μ,σ2), then the random variable M = eW will be distributed lognormally:
f m ( m ; μ , σ 2 ) = 1 m 2 π σ e x p ( − l n ( m − μ ) 2 2 σ 2 ) , z > 0 The parameters μ 0 and σ 0 follow while determining the initial value of the mass variable, M 0 lognormal distribution of m . If the growth of this object with M 0 = m 0 is exponential with growth rate γ , then we can write d m d t = γ m . After time t , the mean of the lognormal distribution would have changed to μ 0 + γ t . However, considering time as a random variable, we can write f ( t ) = δ e x p ( − δ t ) . The closed form of the probability density function of the MLP is as follows:
f ( m ) = α 2 e x p ( α μ 0 + α 2 σ 0 2 2 ) m − ( 1 + α ) e r f c ( 1 2 ( α σ 0 − l n ( m ) − μ 0 σ 0 ) ) , m ∈ [ 0 , ∞ ) where α = δ γ .
Following are the few mathematical properties of the MLP distribution:
The MLP cumulative distribution function ( F ( m ) = ∫ m −∞ f ( t ) d t ) is given by:
F ( m ) = 1 2 e r f c ( − l n ( m ) − μ 0 2 σ 0 ) − 1 2 e x p ( α μ 0 + α 2 σ 0 2 2 ) m − α e r f c ( α σ 0 2 ( α σ 0 − l n ( m ) − μ 0 2 σ 0 ) We can see that as m →0, F ( m ) →1/2 erfc(-ln m -μ0/√2σ0), which is the cumulative distribution function for a lognormal distribution with parameters μ0 and σ0.
The expectation value of M k gives the k th raw moment of M ,
< M k >= ∫ 0 ∞ m k f ( m ) d m This exists if and only if α > k , in which case it becomes:
< M k >= α α − k e x p ( σ 0 2 k 2 2 + μ 0 k ) , α > k which is the k th raw moment of the lognormal distribution with the parameters μ0 and σ0 scaled by α⁄α- k in the limit α→∞. This gives the mean and variance of the MLP distribution:
< M >= α α − 1 e x p ( σ 0 2 2 + μ 0 ) , α > 1 < M 2 >= α α − 2 e x p ( 2 ( σ 0 2 + μ 0 ) ) , α > 2 Var( M ) = ⟨ M 2⟩-(⟨ M ⟩)2 = α exp(σ02 + 2μ0) (exp(σ02)/α-2 - α/(α-2)2), α > 2
The solution to the equation f ′ ( m ) = 0 (equating the slope to zero at the point of maxima)for m gives the mode of the MLP distribution.
f ′ ( m ) = 0 ⇔ K × e r f c ( u ) = e x p ( − u 2 ) where u = (1/√2(ασ0-ln m -μ0/σ0)) and K = (σ0(α+1)√ π⁄2)
Numerical methods are required to solve this transcendental equation. However, noting that if K ≈1 then u = 0 gives us the mode m *:
m ∗ = e x p ( μ 0 + α σ 0 2 ) Random Variate
The lognormal random variate is:
L ( μ , σ ) = e x p ( μ + σ N ( 0 , 1 ) ) where N ( 0 , 1 ) is standard normal random variate. The exponential random variate is :
E ( δ ) = − δ − 1 l n ( R ( 0 , 1 ) ) 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:
M ( μ 0 , σ 0 , α ) = e x p ( μ 0 + σ 0 N ( 0 , 1 ) − α − 1 l n ( R ( 0 , 1 ) ) ) 