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	${\displaystyle \frac{a^{2}z}{((a-1)z+1{)}^2}}$

TheFlory–Schulz distribution is a discreteprobability distribution named afterPaul Flory andGünter Victor Schulz that describes the relative ratios ofpolymers of different length that occur in an ideal step-growthpolymerization process. Theprobability mass function (pmf) for themass fraction of chains of length${\displaystyle k}$ is:$${\displaystyle w_a(k)=a^{2}k(1-a{)}^{k-1}\text{.}}$$

In this equation,k is the number of monomers in the chain,^[1] and0<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 theFischer–Tropsch process that is conceptually related, where it is known asAnderson-Schulz-Flory (ASF)distribution, in that lighterhydrocarbons are converted to heavier hydrocarbons that are desirable as aliquid fuel.

The pmf of this distribution is a solution of the following equation:$${\displaystyle \left\{\begin{array}{l}(a-1)(k+1)w_a(k)+k{w}_a(k+1)=0\text{,}\\ w_a(0)=0\text{,}{w}_a(1)=a^2\text{.}\end{array}\right\}}$$As a probability distribution, one can note that if X and Y are two independent andgeometrically 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 thenegative binomial distribution, with parameters${\displaystyle r=2}$ and${\displaystyle p=a}$.

• Polymers
• Continuous distributions

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