Exponential distribution
Story
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 positive arrival rate, \(\beta\). It is an inverse scale parameter.
Support
The Exponential distribution is supported on the set of nonnegative real numbers.
Probability density function
\[\begin{align}
f(y;\beta) = \beta \,\mathrm{e}^{-\beta y}.
\end{align}\]
Cumulative distribution function
\[\begin{align}
F(y; \beta) = 1 - \mathrm{e}^{-\beta y}.
\end{align}\]
Moments
Mean: \(\displaystyle{\frac{1}{\beta}}\)
Variance: \(\displaystyle{\frac{1}{\beta^2}}\)
Usage
Package |
Syntax |
|---|---|
NumPy |
|
SciPy |
|
Distributions.jl |
|
Stan |
|
Notes
Alternatively, we could parametrize the Exponential distribution in terms of an average time between arrivals of a Poisson process, \(\tau\), as
\[\begin{align}
f(y;\tau) = \frac{1}{\tau}\,\mathrm{e}^{-y/\tau}.
\end{align}\]