Exponential Distribution Reference Cheat Sheet

Key Facts

• If $X\sim \operatorname{Exp}(\lambda)$, then the probability density function is $f(x)=\lambda e^{-\lambda x}$ for $x\ge 0$ and $f(x)=0$ for $x<0$.
• The cumulative distribution function is $F(x)=P(X\le x)=1-e^{-\lambda x}$ for $x\ge 0$.
• The survival function is $P(X>x)=1-F(x)=e^{-\lambda x}$ for $x\ge 0$.
• The mean waiting time is $E[X]=\frac{1}{\lambda}$, so larger $\lambda$ means shorter average waiting time.
• The variance and standard deviation are $\operatorname{Var}(X)=\frac{1}{\lambda^2}$ and $\sigma=\frac{1}{\lambda}$.
• The memoryless property is $P(X>s+t\mid X>s)=P(X>t)=e^{-\lambda t}$ for $s,t\ge 0$.
• The median is $m=\frac{\ln 2}{\lambda}$ because $P(X\le m)=0.5$.
• If events follow a Poisson process with rate $\lambda$, then the waiting time until the next event follows $\operatorname{Exp}(\lambda)$.

Vocabulary

Exponential distribution

A continuous probability distribution used to model waiting time until the next event in a constant-rate process.

Rate parameter

The parameter $\lambda$ represents the average number of events per unit time and must satisfy $\lambda>0$.

Probability density function

The function $f(x)=\lambda e^{-\lambda x}$ describes relative likelihood for possible waiting times $x\ge 0$.

Cumulative distribution function

The function $F(x)=1-e^{-\lambda x}$ gives the probability that the waiting time is at most $x$.

Survival function

The function $S(x)=P(X>x)=e^{-\lambda x}$ gives the probability that the waiting time exceeds $x$.

Memoryless property

The rule $P(X>s+t\mid X>s)=P(X>t)$ means that the remaining waiting time does not depend on time already waited.