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The graph of a probability mass function. All the values of this function must be non-negative and sum up to 1. In probability and statistics, a probability mass function (sometimes called probability function or frequency function [1]) is a function that gives the probability that a discrete random variable is exactly equal to some value. [2]
In statistics, especially in Bayesian statistics, the kernel of a probability density function (pdf) or probability mass function (pmf) is the form of the pdf or pmf in which any factors that are not functions of any of the variables in the domain are omitted. [1] Note that such factors may well be functions of the parameters of the
It is equivalent to, and sometimes called, the z-transform of the probability mass function. Other generating functions of random variables include the moment-generating function, the characteristic function and the cumulant generating function.
The two-parameter generalization of the original Yule distribution replaces the beta function with an incomplete beta function.The probability mass function of the generalized Yule–Simon(ρ, α) distribution is defined as
This leads directly to the probability mass function of a Log(p)-distributed random variable: = for k ≥ 1, and where 0 < p < 1. Because of the identity above, the distribution is properly normalized. The cumulative distribution function is
The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density functions respectively.
The probability mass function of a Poisson-distributed random variable with mean μ is given by (;) =!.for (and zero otherwise). The Skellam probability mass function for the difference of two independent counts = is the convolution of two Poisson distributions: (Skellam, 1946)
The probability mass function (pmf) for the mass fraction of chains of length is: () = (). 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.
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