Exponential Distribution

#probability #distribution #continuous_distribution

Gist

A distribution known for its use in waiting time.

Mathematics

This is a continuous probability distribution.

The probability density function is given by:

f_{X} (x) = λ e^{- λ x} $ $ A n d t h e c u m u l a t i v e d e n s i t y f u n c t i o n i s : $ $ F_{X} (x) = 1 - e^{- λ x}

Properties

Mean: $\frac{1}{x}$

Derivation

\begin{aligned} E [X] & = \int_{0}^{\infty} x λ e^{- λ x} d x \\ = {[- x e^{- λ x}]}_{0}^{\infty} + \int_{0}^{\infty} e^{- λ x} d x \\ = 0 + {[- \frac{1}{λ} e^{- λ x}]}_{0}^{\infty} \\ = \frac{1}{λ} \end{aligned}

Variance: $\frac{1}{λ^{2}}$

Memoryless

The memoryless property states:

P (X > t + s | X > s) = P (X > t),

Given that the event hasn't occurred by time $s$ , the probability of the event happening later ( $t + s$ ) is the same as the probability that it has not occurred by time $t$ from the start. It does not matter how long you already waited for. The distribution of the remaining wait time is identical to the wait you faced at time zero.

Uses

In compartmental modeling (like Model - SIR), the waiting time of individuals is described with the exponential distribution.
A possible choice for Survival analysis

Relationship

A continuous form of the Geometric distribution.
The sum of $n$ exponential distributions is the equivalent of the Gamma Distribution which is useful for Delay Distribution

Other notes

Interestingly, in 1958, it was considered not a useful distribution for application?