About 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 $0 \leq F_{X} (x) \leq 1$ 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. Only the discrete-uniform type is supported in Isight.

The following nomenclature is used throughout this section:

$X$	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