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

$p\left(x\right) = p_{0}\left(x-L\right)$

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

$p_{0}\left(x\right) = 0\quad x < a \textrm{ or } x > b$

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 evaluate 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.

$p\left(x_{k}\right)=P\left[X=x_{k}\right]$

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

$f\left(x\right)=\sum_{k}p\left(x_{k}\right)\delta\left(x-x_{k}\right)$

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

$F\left(x\right)=P\left[X\leq x\right]=\sum_{x_{k}\leq x}p\left(x_{k}\right)$

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

$F\left(x_{k}\right)-F\left(x_{k-1}\right)=p\left(x_{k}\right)$

Survival Function¶

The survival function is just

$S\left(x\right)=1-F\left(x\right)=P\left[X>k\right]$

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

$G\left(q\right)=F^{-1}\left(q\right)$

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

$Z\left(\alpha\right)=S^{-1}\left(\alpha\right)=G\left(1-\alpha\right)$

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

$h\left(x_{k}\right)=\frac{p\left(x_{k}\right)}{1-F\left(x_{k}\right)}$

and

$H\left(x\right)=\sum_{x_{k}\leq x}h\left(x_{k}\right)=\sum_{x_{k}\leq x}\frac{F\left(x_{k}\right)-F\left(x_{k-1}\right)}{1-F\left(x_{k}\right)}.$

Moments¶

Non-central moments are defined using the PDF

$\mu_{m}^{\prime}=E\left[X^{m}\right]=\sum_{k}x_{k}^{m}p\left(x_{k}\right).$

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

$\begin{eqnarray*} \mu_{m}=E\left[\left(X-\mu\right)^{m}\right] & = & \sum_{k}\left(x_{k}-\mu\right)^{m}p\left(x_{k}\right)\\ & = & \sum_{k=0}^{m}\left(-1\right)^{m-k}\left(\begin{array}{c} m\\ k\end{array}\right)\mu^{m-k}\mu_{k}^{\prime}\end{eqnarray*}$

The mean is the first moment

$\mu=\mu_{1}^{\prime}=E\left[X\right]=\sum_{k}x_{k}p\left(x_{k}\right)$

the variance is the second central moment

$\mu_{2}=E\left[\left(X-\mu\right)^{2}\right]=\sum_{x_{k}}x_{k}^{2}p\left(x_{k}\right)-\mu^{2}.$

Skewness is defined as

$\gamma_{1}=\frac{\mu_{3}}{\mu_{2}^{3/2}}$

while (Fisher) kurtosis is

$\gamma_{2}=\frac{\mu_{4}}{\mu_{2}^{2}}-3,$

so that a normal distribution has a kurtosis of zero.

Moment generating function¶

The moment generating function is defined as

$M_{X}\left(t\right)=E\left[e^{Xt}\right]=\sum_{x_{k}}e^{x_{k}t}p\left(x_{k}\right)$

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

$f\left(\mathbf{k};\boldsymbol{\theta}\right)=\prod_{i=1}^{N}f_{i}\left(k_{i};\boldsymbol{\theta}\right).$

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:

$\begin{eqnarray*} \hat{\boldsymbol{\theta}} & = & \arg\max_{\boldsymbol{\theta}}f\left(\mathbf{k};\boldsymbol{\theta}\right)\\ & = & \arg\min_{\boldsymbol{\theta}}l_{\mathbf{k}}\left(\boldsymbol{\theta}\right).\end{eqnarray*}$

Where

$\begin{eqnarray*} l_{\mathbf{k}}\left(\boldsymbol{\theta}\right) & = & -\sum_{i=1}^{N}\log f\left(k_{i};\boldsymbol{\theta}\right)\\ & = & -N\overline{\log f\left(k_{i};\boldsymbol{\theta}\right)}\end{eqnarray*}$

Standard notation for mean¶

We will use

$\overline{y\left(\mathbf{x}\right)}=\frac{1}{N}\sum_{i=1}^{N}y\left(x_{i}\right)$

where $N$ should be clear from context.

Combinations¶

Note that

$k!=k\cdot\left(k-1\right)\cdot\left(k-2\right)\cdot\cdots\cdot1=\Gamma\left(k+1\right)$

and has special cases of

$\begin{eqnarray*} 0! & \equiv & 1\\ k! & \equiv & 0\quad k<0\end{eqnarray*}$

and

$\left(\begin{array}{c} n\\ k\end{array}\right)=\frac{n!}{\left(n-k\right)!k!}.$

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