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 interarrival 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
Moments
Mean: \(\displaystyle{\frac{1}{\beta}}\)
Variance: \(\displaystyle{\frac{1}{\beta^2}}\)
Usage
Package 
Syntax 

NumPy 

SciPy 

Stan 

Notes
Alternatively, we could parametrize the Exponential distribution in terms of an average time between arrivals of a Poisson process, \(\tau\), as
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 thatscipy.stats
requires it. Furthermore, thescipy.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 \(\tau\).