Exponential distribution

Story

Rare events occur with a rate \(\beta\) per unit time. There is no “memory” of previous events; i.e., that rate is independent of time. A process that generates such events is called a Poisson process. The occurrence of a rare event in this context is referred to as an arrival. The inter-arrival time of a Poisson process is Exponentially distributed.

Example

The time between conformational switches in a protein is Exponentially distributed (under simple mass action kinetics).

Parameters

The single parameter is the average arrival rate, \(\beta\).

Support

The Exponential distribution is supported on the set of nonnegative real numbers.

Probability density function

\[\begin{align} f(y;\beta) = \beta \,\mathrm{e}^{-\beta y}. \end{align}\]

Moments

Mean: \(\displaystyle{\frac{1}{\beta}}\)

Variance: \(\displaystyle{\frac{1}{\beta^2}}\)

Usage

Package

Syntax

NumPy

rg.exponential(1/beta)

SciPy

scipy.stats.expon(loc=0, scale=1/beta)

Stan

exponential(beta)

Notes

  • Alternatively, we could parametrize the Exponential distribution in terms of an average time between arrivals of a Poisson process, \(\tau\), as

\[\begin{align} f(y;\tau) = \frac{1}{\tau}\,\mathrm{e}^{-y/\tau}. \end{align}\]

  • The implementation in the scipy.stats module also has a location parameter, which shifts the distribution left and right. For our purposes, you can ignore that parameter, but be aware that scipy.stats requires it. Furthermore, the scipy.stats Exponential distribution is parametrized in terms of the interarrival time \(\tau\) and not the arrival rate \(\beta\).

  • NumPy’s rg.exponential() function does not need nor accept a location parameter. It is also parametrized in terms of \(\tau\).

PDF and CDF plots