Flory–Schulz distribution

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

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 $k$ is: $$w_a(k)=a^{2}k(1-a{)}^{k-1}\text{.}$$

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

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: $$\left\{\begin{array}{c}(a-1)(k+1)w_a(k)+k{w}_a(k+1)=0\text{,}\\ w_a(0)=0\text{,}{w}_a(1)=a^2\text{.}\end{array}\right\}$$As a probability distribution, one can note that if X and Y are two independent and geometrically distributed random variables with parameter $a$ taking values in $\{0,1,\cdots \}$, then$$w_a(k)=\mathbb{P}(X+Y+1=k)$$This in turn means that the Flory-Schulz distribution is a shifted version of the negative binomial distribution, with parameters $r=2$ and $p=a$.