About Probability Distributions

For a given random variable $X$ , the probability that $X$ will take on a value $X$ is defined by the probability density function for that random variable:

f_{X} (x) = \Pr [X = x],

where $f_{X} (x) \geq 0$ for all $x$ . The probability that the random variable $X$ will take on a value less than a specified threshold value $x$ is defined by the distribution function for that random variable, often also termed the cumulative distribution function:

F_{X} (x) = \Pr [X \leq x],

where $0 \leq F_{X} (x) \leq 1$ for all $x$ . For a continuous random variable $X$ , the probability density function, $f_{X} (x)$ , and cumulative distribution function, $F_{X} (x)$ , are related as follows:

\begin{array}{l} F_{X} (x) = \int_{- \infty}^{x} f (t) d t \\ f_{X} (x) = \frac{d (F_{X} (x))}{d x} . \end{array}

The probability density and cumulative distribution functions for a given probability distribution are generally defined as a function of one or more distribution parameters that define the location, shape, or dispersion of the distribution. The following are given for each distribution type:

the probability density and cumulative distributions, and
the translation between the distribution parameters and the mean and standard deviation statistics of a random variable.

Note: The integral in the previous equation becomes a summation for discrete random variables, where the summation is taken over the discrete probability values associated with the set of values for the random variable. Process Composer supports only the discrete-uniform type.

The following nomenclature is used in the descriptions of the probability distribution types:


$X$	random variable
$f_{X} (x)$	probability density function
$F_{X} (x)$	distribution function
$μ$	mean
$σ$	standard deviation
$γ$	Euler’s constant (Gumbel)
$Γ$	gamma function