Discrete Statistical Distributions#

Overview#

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] .

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

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

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