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For computing the PMF, a DFT algorithm or a recursive algorithm can be specified to compute the exact PMF, and approximation methods using the normal and Poisson distribution can also be specified. poibin - Python implementation - can compute the PMF and CDF, uses the DFT method described in the paper for doing so.
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]
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 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.
Here F X is the cumulative distribution function of X, f X is the corresponding probability density function, Q X (p) is the corresponding inverse cumulative distribution function also called the quantile function, [2] and the integrals are of the Riemann–Stieltjes kind.
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. [2]
Converting a single distribution to a location–scale family [ edit ] The following shows how to implement a location–scale family in a statistical package or programming environment where only functions for the "standard" version of a distribution are available.
Since the ZTP is a truncated distribution with the truncation stipulated as k > 0, one can derive the probability mass function g(k;λ) from a standard Poisson distribution f(k;λ) as follows: [4] g ( k ; λ ) = P ( X = k ∣ X > 0 ) = f ( k ; λ ) 1 − f ( 0 ; λ ) = λ k e − λ k !