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The MLP (Modified Lognormal Power-Law) function is a functional form of the Initial Mass Function(IMF). It is a three parameter function that can be used to model data that have characteristics of lognormal distribution and a power-law behavior. Unlike the other functional forms of the IMF, the initial mass function is a single function with no joining conditions.

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 (CITATION REQUIRED):

$$f$$$m$($$m$$;μ,σ2) = $1⁄$m$√(2π)σ$ exp(-$(ln$m$ - μ)^{2}⁄2σ^{2}$), $$z$$ > 0

The parameters μ0 and σ0 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 $$dm$⁄$dt$$=γ$$m$$. After time $$t$$, the mean of the lognormal distribution would have changed to μ0+γ$$t$$. However, considering time as a random variable, we can write $$f(t)$$ = δ exp(-δ$$t$$). The closed form of the probability density function of the MLP is as follows:

$$f(m)$$ = $α⁄2$ exp(αμ0+$α^{2}σ_{0}^{2}⁄2$) m-(1+α) erfc($1⁄√2$(ασ0-$ln$m$-μ_{0}⁄σ_{0}$)), $$m$$ ∈ [0,∞)

where α = δ/γ.

Mathematical Properties of the MLP distribution
Following are the few mathematical properties of the MLP distribution:

Cumulative Distribution
The MLP cumulative distribution function ($$F(m)$$ = ∫$m$-∞ $$f(t)dt$$) is given by:

$$F(m)$$ = $1⁄2$ erfc(-$ln$m$-μ_{0}⁄√2σ_{0}$) - $1⁄2$ exp(αμ0+$α^{2}σ_{0}^{2}⁄2$) m-α erfc($ασ_{0}⁄√2$(ασ0-$ln$m$-μ_{0}⁄√2σ_{0}$))

We can see that as $$m$$→0, $$F(m)$$→$1⁄2$ erfc(-$ln$m$-μ_{0}⁄√2σ_{0}$), which is the cumulative distribution function for a lognormal distribution with parameters μ0 and σ0.

Mean, Variance, Raw Moments, Mode
The expectation value of $$M$$k gives the $$k$$th raw moment of $$M$$,

⟨$$M$$k⟩ = ∫∞0 $$m$$k $$f(m)dm$$

This exists if and only if α > $$k$$, in which case it becomes:

⟨$$M$$k⟩ = $α⁄α-$k$$ exp($σ_{0}^{2}$k$^{2}⁄2$ + μ0$$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:

⟨$$M$$⟩ = $α⁄α-1$ exp($σ_{0}^{2}⁄2$ + μ0), α > 1

⟨$$M$$2⟩ = $α⁄α-2$ exp(2(σ02 + μ0)), α > 2

Var($$M$$) = ⟨$$M$$2⟩-(⟨$$M$$⟩)2 = α exp(σ02 + 2μ0) ($exp(σ_{0}^{2})⁄α-2$ - $α⁄(α-2)^{2}$), α > 2