# Discrete Statistical Distributions¶

Discrete random variables take on only a countable number of values. The commonly used distributions are included in SciPy and described in this document. Each discrete distribution can take one extra integer parameter: \(L.\) The relationship between the general distribution \(p\) and the standard distribution \(p_{0}\) is

which allows for shifting of the input. When a distribution generator is initialized, the discrete distribution can either specify the beginning and ending (integer) values \(a\) and \(b\) which must be such that

in which case, it is assumed that the pdf function is specified on the integers \(a+mk\leq b\) where \(k\) is a non-negative integer ( \(0,1,2,\ldots\) ) and \(m\) is a positive integer multiplier. Alternatively, the two lists \(x_{k}\) and \(p\left(x_{k}\right)\) can be provided directly in which case a dictionary is set up internally to evaulate probabilities and generate random variates.

## Probability Mass Function (PMF)¶

The probability mass function of a random variable X is defined as the probability that the random variable takes on a particular value.

This is also sometimes called the probability density function, although technically

is the probability density function for a discrete distribution [1] .

[1] | XXX: Unknown layout Plain Layout: Note that we will be using \(p\) to represent the probability mass function and a parameter (a XXX: probability). The usage should be obvious from context. |

## Cumulative Distribution Function (CDF)¶

The cumulative distribution function is

and is also useful to be able to compute. Note that

## Survival Function¶

The survival function is just

the probability that the random variable is strictly larger than \(k\) .

## Percent Point Function (Inverse CDF)¶

The percent point function is the inverse of the cumulative distribution function and is

for discrete distributions, this must be modified for cases where there is no \(x_{k}\) such that \(F\left(x_{k}\right)=q.\) In these cases we choose \(G\left(q\right)\) to be the smallest value \(x_{k}=G\left(q\right)\) for which \(F\left(x_{k}\right)\geq q\) . If \(q=0\) then we define \(G\left(0\right)=a-1\) . This definition allows random variates to be defined in the same way as with continuous rv’s using the inverse cdf on a uniform distribution to generate random variates.

## Inverse survival function¶

The inverse survival function is the inverse of the survival function

and is thus the smallest non-negative integer \(k\) for which \(F\left(k\right)\geq1-\alpha\) or the smallest non-negative integer \(k\) for which \(S\left(k\right)\leq\alpha.\)

## Hazard functions¶

If desired, the hazard function and the cumulative hazard function could be defined as

and

## Moments¶

Non-central moments are defined using the PDF

Central moments are computed similarly \(\mu=\mu_{1}^{\prime}\)

The mean is the first moment

the variance is the second central moment

Skewness is defined as

while (Fisher) kurtosis is

so that a normal distribution has a kurtosis of zero.

## Moment generating function¶

The moment generating function is defined as

Moments are found as the derivatives of the moment generating function evaluated at \(0.\)

## Fitting data¶

To fit data to a distribution, maximizing the likelihood function is common. Alternatively, some distributions have well-known minimum variance unbiased estimators. These will be chosen by default, but the likelihood function will always be available for minimizing.

If \(f_{i}\left(k;\boldsymbol{\theta}\right)\) is the PDF of a random-variable where \(\boldsymbol{\theta}\) is a vector of parameters ( *e.g.* \(L\) and \(S\) ), then for a collection of \(N\) independent samples from this distribution, the joint distribution the
random vector \(\mathbf{k}\) is

The maximum likelihood estimate of the parameters \(\boldsymbol{\theta}\) are the parameters which maximize this function with \(\mathbf{x}\) fixed and given by the data:

Where

## Standard notation for mean¶

We will use

where \(N\) should be clear from context.

## Combinations¶

Note that

and has special cases of

and

If \(n<0\) or \(k<0\) or \(k>n\) we define \(\left(\begin{array}{c} n\\ k\end{array}\right)=0\)

## Discrete Distributions in `scipy.stats`¶

- Bernoulli Distribution
- Binomial Distribution
- Boltzmann (truncated Planck) Distribution
- Planck (discrete exponential) Distribution
- Poisson Distribution
- Geometric Distribution
- Negative Binomial Distribution
- Hypergeometric Distribution
- Zipf (Zeta) Distribution
- Logarithmic (Log-Series, Series) Distribution
- Discrete Uniform (randint) Distribution
- Discrete Laplacian Distribution