Flory–Schulz distribution

The Flory–Schulz distribution is a discrete probability distribution named after Paul Flory and Günter Victor Schulz that describes the relative ratios of polymers of different length that occur in an ideal step-growth polymerization process. The probability mass function (pmf) for the mass fraction of chains of length ${\displaystyle k}$ is: $${\displaystyle w_{a}(k)=a^{2}k(1-a)^{k-1}{\text{.}}}$$

Flory–Schulz distribution
Probability mass function
Parameters	0 < a < 1 (real)
Support	k ∈ { 1, 2, 3, ... }
PMF	${\displaystyle a^{2}k(1-a)^{k-1}}$
CDF	${\displaystyle 1-(1-a)^{k}(1+ak)}$
Mean	${\displaystyle {\frac {2}{a}}-1}$
Median	${\displaystyle {\frac {W\left({\frac {(1-a)^{\frac {1}{a}}\log(1-a)}{2a}}\right)}{\log(1-a)}}-{\frac {1}{a}}}$
Mode	${\displaystyle -{\frac {1}{\log(1-a)}}}$
Variance	${\displaystyle {\frac {2-2a}{a^{2}}}}$
Skewness	${\displaystyle {\frac {2-a}{\sqrt {2-2a}}}}$
Excess kurtosis	${\displaystyle {\frac {(a-6)a+6}{2-2a}}}$
MGF	${\displaystyle {\frac {a^{2}e^{t}}{\left((a-1)e^{t}+1\right)^{2}}}}$
CF	${\displaystyle {\frac {a^{2}e^{it}}{\left(1+(a-1)e^{it}\right)^{2}}}}$
PGF	${\displaystyle {\frac {a^{2}z}{((a-1)z+1)^{2}}}}$

In this equation, k is the number of monomers in the chain,^[1] and 0<a<1 is an empirically determined constant related to the fraction of unreacted monomer remaining.^[2]

The form of this distribution implies is that shorter polymers are favored over longer ones — the chain length is geometrically distributed. Apart from polymerization processes, this distribution is also relevant to the Fischer–Tropsch process that is conceptually related, where it is known as Anderson-Schulz-Flory (ASF) distribution, in that lighter hydrocarbons are converted to heavier hydrocarbons that are desirable as a liquid fuel.

The pmf of this distribution is a solution of the following equation: $${\displaystyle \left\{{\begin{array}{l}(a-1)(k+1)w_{a}(k)+kw_{a}(k+1)=0{\text{,}}\\[10pt]w_{a}(0)=0{\text{,}}w_{a}(1)=a^{2}{\text{.}}\end{array}}\right\}}$$As a probability distribution, one can note that if X and Y are two independent and geometrically distributed random variables with parameter ${\displaystyle a}$ taking values in ${\displaystyle \{0,1,\cdots \}}$, then$${\displaystyle w_{a}(k)=\mathbb {P} \left(X+Y+1=k\right)}$$This in turn means that the Flory-Schulz distribution is a shifted version of the negative binomial distribution, with parameters ${\displaystyle r=2}$ and ${\displaystyle p=a}$.