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# 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: The relationship between the general distribution and the standard distribution 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 and which must be such that

in which case, it is assumed that the pdf function is specified on the integers where is a non-negative integer ( ) and is a positive integer multiplier. Alternatively, the two lists and 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 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 .

## 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 such that In these cases we choose to be the smallest value for which . If then we define . 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 for which or the smallest non-negative integer for which

## 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

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 funtion is defined as

Moments are found as the derivatives of the moment generating function evaluated at

## 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 is the PDF of a random-variable where is a vector of parameters ( e.g. and ), then for a collection of independent samples from this distribution, the joint distribution the random vector is

The maximum likelihood estimate of the parameters are the parameters which maximize this function with fixed and given by the data:

Where

## Standard notation for mean

We will use

where should be clear from context.

## Combinations

Note that

and has special cases of

and

If or or we define

### Bernoulli

A Bernoulli random variable of parameter takes one of only two values or . The probability of success ( ) is , and the probability of failure ( ) is It can be thought of as a binomial random variable with . The PMF is for and

### Binomial

A binomial random variable with parameters can be described as the sum of independent Bernoulli random variables of parameter

Therefore, this random variable counts the number of successes in independent trials of a random experiment where the probability of success is

where the incomplete beta integral is

Now

### Boltzmann (truncated Planck)

Define

### Planck (discrete exponential)

Named Planck because of its relationship to the black-body problem he solved.

### Poisson

The Poisson random variable counts the number of successes in independent Bernoulli trials in the limit as and where the probability of success in each trial is and is a constant. It can be used to approximate the Binomial random variable or in it’s own right to count the number of events that occur in the interval for a process satisfying certain “sparsity “constraints. The functions are

### Geometric

The geometric random variable with parameter can be defined as the number of trials required to obtain a success where the probability of success on each trial is . Thus,

### Negative Binomial

The negative binomial random variable with parameters and can be defined as the number of extra independent trials (beyond ) required to accumulate a total of successes where the probability of a success on each trial is Equivalently, this random variable is the number of failures encoutered while accumulating successes during independent trials of an experiment that succeeds with probability Thus,

Recall that is the incomplete beta integral.

### Hypergeometric

The hypergeometric random variable with parameters counts the number of “good “objects in a sample of size chosen without replacement from a population of objects where is the number of “good “objects in the total population.

where (defining )

### Zipf (Zeta)

A random variable has the zeta distribution (also called the zipf distribution) with parameter if it’s probability mass function is given by

where

is the Riemann zeta function. Other functions of this distribution are

where and is the polylogarithm function of defined as

### Logarithmic (Log-Series, Series)

The logarimthic distribution with parameter has a probability mass function with terms proportional to the Taylor series expansion of

where

is the Lerch Transcendent. Also define

Thus,

### Discrete Uniform (randint)

The discrete uniform distribution with parameters constructs a random variable that has an equal probability of being any one of the integers in the half-open range If is not given it is assumed to be zero and the only parameter is Therefore,

### Discrete Laplacian

Defined over all integers for

Thus,

where is the polylogarithm function of order evaluated at

### Discrete Gaussian*

Defined for all and and

where

where and with a minimum less than 2 near

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