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	${\frac {a^{2}e^{t}}{\left((a-1)e^{t}+1\right)^{2}}}$
CF	${\frac {a^{2}e^{it}}{\left(1+(a-1)e^{it}\right)^{2}}}$
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,^[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, 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}{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\}

References

^ Flory, Paul J. (October 1936). "Molecular Size Distribution in Linear Condensation Polymers". Journal of the American Chemical Society. 58 (10): 1877–1885. doi:10.1021/ja01301a016. ISSN 0002-7863.
^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "most probable distribution". doi:10.1351/goldbook.M04035

Probability distributions (list)

Discrete
univariate

with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric negative Poisson binomial Rademacher soliton discrete uniform Zipf Zipf–Mandelbrot
with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin geometric logarithmic mixed Poisson negative binomial Panjer parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous
univariate

supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular continuous Bernoulli Irwin–Hall Kumaraswamy logit-normal noncentral beta PERT raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi chi-squared noncentral inverse scaled Dagum Davis Erlang hyper exponential hyperexponential hypoexponential logarithmic F noncentral folded normal Fréchet gamma generalized inverse gamma/Gompertz Gompertz shifted half-logistic half-normal Hotelling's T-squared inverse Gaussian generalized Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal log-t Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto phase-type Poly-Weibull Rayleigh relativistic Breit–Wigner Rice truncated normal type-2 Gumbel Weibull discrete Wilks's lambda
supported on the whole real line	Cauchy exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t Tracy–Widom variance-gamma Voigt
with support whose type varies	generalized chi-squared generalized extreme value generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda