Statistical Distributions

Statistical Distributions#

For a general introduction to probability distributions, see: Wikipedia [1228], Wikipedia [1220], Wikipedia [1218], Wikipedia [1235].

The design of this library has been influenced by Bristow et al. [166], Witkovsky [1609].

See also Wikipedia [1215] and Abramowitz and Stegun. [4] (from which the following introductory text has been taken, with small edits).

A real-valued function \(F(x)\) is termed a (univariate) cumulative distribution function (cdf) or simply distribution function if

  1. \(F(x)\) is non-decreasing, i.e. \(F(x_1) \le F(x_2)\) for \(x_1 \le x_2\).

  2. \(F(x)\) is everywhere continuous from the right, i.e. \(F(x) = \lim \limits_{\epsilon \to 0+} F(x+\epsilon)\)

  3. \(F(-\infty) = 0, F(\infty) = 1\).

The function \(F(x)\) signifies the probability of the event “\(X \le x\)”, i.e. Pr \(\{X \le x \} = F(x),\) where \(X\) is a random variable, and thus describes the cdf of \(X.\) The two principle types of distribution functions are termed discrete and continuous.

Discrete distributions: Discrete distributions are characterized by the random variable \(X\) taking on an enumerable number of values \(\ldots x_{-1}, x_0, x_1, \ldots\) with point probabilities \(p_n = \text{Pr} \{X = x_n \} \ge 0,\) which only need to be subject to the restriction \(\sum_n p_n =1.\) The corresponding distribution function can then be written as

\[F(x) = \text{Pr} \{X \le x \} = \sum_{x_n \le x} p_n,\]

where the summation is over all values of x for which \(x_n \le x\). The set \(\{ x_n \}\) of values for which \(p_n > 0\) is termed the domain of he random variable \(X.\) A discrete distribution of a random variable is called a lattice distribution if there exists numbers \(a\) and \(b \ne 0\) such that every possible value of \(X\) can be represented in the form \(a+bn\) where \(n\) takes only integral values.

Continuous distributions: Continuous distributions are characterized by \(F(x)\) being continuous. If \(F(x)\) is absolutely continuous then \(F(x)\) possesses a derivative \(F'(x) = f(x)\) and the cdf can be written as

\[F(x) = \text{Pr} \{X \le x \} = \int_{-\infty}^x f(t) \mathrm{d} t.\]

The derivative \(f(x)\) is termed the probability density function (pdf), and the values of \(x\) for which \(f(x) > 0\) make up the domain of the variable \(X\).

Statistical Distributions