stdlib_stats_distribution_exponential.md

title
stats_distribution_exponential

Statistical Distributions -- Exponential Distribution Module

[TOC]

rvs_exp - exponential distribution random variates

Status

Experimental

Description

An exponential distribution is the distribution of time between events in a Poisson point process. The inverse scale parameter lambda specifies the average time between events ((\lambda)), also called the rate of events.

Without argument, the function returns a random sample from the standard exponential distribution (E(\lambda=1)).

With a single argument, the function returns a random sample from the exponential distribution (E(\lambda=\text{lambda})). For complex arguments, the real and imaginary parts are sampled independently of each other.

With two arguments, the function returns a rank-1 array of exponentially distributed random variates.

@note The algorithm used for generating exponential random variates is fundamentally limited to double precision.¹

Syntax

result = [[stdlib_stats_distribution_exponential(module):rvs_exp(interface)]]([lambda] [[, array_size]])

Class

Elemental function

Arguments

lambda: optional argument has intent(in) and is a scalar of type real or complex. If lambda is real, its value must be positive. If lambda is complex, both the real and imaginary components must be positive.

array_size: optional argument has intent(in) and is a scalar of type integer with default kind.

Return value

The result is a scalar or rank-1 array with a size of array_size, and the same type as lambda. If lambda is non-positive, the result is NaN.

Example

{!example/stats_distribution_exponential/example_exponential_rvs.f90!}

pdf_exp - exponential distribution probability density function

Status

Experimental

Description

The probability density function (pdf) of the single real variable exponential distribution is:

$$f(x)=\begin{cases} \lambda e^{-\lambda x} &x\geqslant 0 \\ 0 &x< 0\end{cases}$$

For a complex variable (z=(x + y i)) with independent real (x) and imaginary (y) parts, the joint probability density function is the product of the corresponding real and imaginary marginal pdfs:²

$$f(x+\mathit{i}y)=f(x)f(y)=\begin{cases} \lambda_{x} \lambda_{y} e^{-(\lambda_{x} x + \lambda_{y} y)} &x\geqslant 0, y\geqslant 0 \\ 0 &\text{otherwise}\end{cases}$$

Syntax

result = [[stdlib_stats_distribution_exponential(module):pdf_exp(interface)]](x, lambda)

Class

Elemental function

Arguments

x: has intent(in) and is a scalar of type real or complex.

lambda: has intent(in) and is a scalar of type real or complex. If lambda is real, its value must be positive. If lambda is complex, both the real and imaginary components must be positive.

All arguments must have the same type.

Return value

The result is a scalar or an array, with a shape conformable to the arguments, and the same type as the input arguments. If lambda is non-positive, the result is NaN.

Example

{!example/stats_distribution_exponential/example_exponential_pdf.f90!}

cdf_exp - exponential cumulative distribution function

Status

Experimental

Description

Cumulative distribution function (cdf) of the single real variable exponential distribution:

$$F(x)=\begin{cases}1 - e^{-\lambda x} &x\geqslant 0 \\ 0 &x< 0\end{cases}$$

For a complex variable (z=(x + y i)) with independent real (x) and imaginary (y) parts, the joint cumulative distribution function is the product of corresponding real and imaginary marginal cdfs:²

$$F(x+\mathit{i}y)=F(x)F(y)=\begin{cases} (1 - e^{-\lambda_{x} x})(1 - e^{-\lambda_{y} y}) &x\geqslant 0, ;; y\geqslant 0 \\ 0 & \text{otherwise} \end{cases}$$

Syntax

result = [[stdlib_stats_distribution_exponential(module):cdf_exp(interface)]](x, lambda)

Class

Elemental function

Arguments

x: has intent(in) and is a scalar of type real or complex.

lambda: has intent(in) and is a scalar of type real or complex. If lambda is real, its value must be positive. If lambda is complex, both the real and imaginary components must be positive.

All arguments must have the same type.

Return value

The result is a scalar or an array, with a shape conformable to the arguments, and the same type as the input arguments. If lambda is non-positive, the result is NaN.

Example

{!example/stats_distribution_exponential/example_exponential_cdf.f90!}

Footnotes

1. Marsaglia, George, and Wai Wan Tsang. "The ziggurat method for generating random variables." Journal of statistical software 5 (2000): 1-7. ↩

2. Miller, Scott, and Donald Childers. Probability and random processes: With applications to signal processing and communications. Academic Press, 2012 (p. 197). ↩ ↩²