Exponential Distribution

Compact study note.

Summary

The exponential distribution models waiting time to the next event in one constant-rate Poisson process. It is right-skewed and memoryless, not symmetric.^[1]

Prerequisites

• Poisson Distribution
• Continuous Random Variable

Definition

$X\sim \mathrm{Exponential}(\lambda )$ uses the rate parameterization.

Notation and Assumptions

$\lambda >0$ is a rate per unit of time or space. Support is nonnegative.

Parameters

$\lambda >0$ .

Support

$[0,\mathrm{\infty })$ .

PMF or PDF

$f_X(x)=\lambda e^{-\lambda x}$ for $x\ge 0$ , and $0$ otherwise.

CDF

$F_X(x)=1-e^{-\lambda x}$ for $x\ge 0$ .

Moments

$E[X]=1/\lambda$ , $\mathrm{Var}(X)=1/{\lambda }^2$ , and $M_X(t)=\lambda /(\lambda -t)$ for $t<\lambda$ .

Essential Result

Memorylessness: $P(X>s+t\mid X>s)=P(X>t)$ for $s,t\ge 0$ .

Small Example

If calls arrive at rate $2$ per minute, the chance of waiting more than one minute is $P(X>1)=e^{-2}$ .

Common Mistakes

• Mixing rate $\lambda$ with scale $\theta =1/\lambda$ .
• Calling the exponential distribution symmetric.

Connections

• Poisson Distribution
• Gamma Distribution
• Weibull Distribution

References