Exponential Distribution

Definition

Let $X$ be a discrete random variable with

Parameters

• $\lambda$: waiting time between two successive events in a Poisson process with the average $\lambda$.

A random variable is said to follow a exponential distribution if the probability is described by the following

Probability Density Function

$$\begin{array}{rrcl}{p}_X:& \mathbb{R}& \to & \mathbb{R}\\ & x& \to & \{\begin{array}{rcl}\lambda e^{-\lambda x},& \text{if}& x>0\\ 0,& \text{if}& x\le 0.\end{array}\end{array}$$

Moments

• Expected value: $E[X]=1/\lambda$,
• Variance: $\text{Var}(X)=1/{\lambda }^2$.

Uncertainty Estimation

Todo

Relationship with Other Distributions

Gamma Distribution

When $\alpha =1$, the Gamma distribution reduces to the exponential distribution with parameter $\beta =\lambda$.

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