Exponential Distribution — Time Between Events
Foundations of Statistics
Modeling Waiting Times and Survival
The exponential distribution is the mathematical backbone for modeling time between random events, from customer arrivals to equipment failures. Its unique memoryless property makes it indispensable for reliability engineering and queueing theory.
- Reliability Engineering — Predicting time until component failure in aerospace and electronics
- Telecommunications — Modeling call arrivals and service times in network traffic analysis
- Healthcare — Estimating waiting times between patient arrivals in emergency departments
The exponential distribution tells us how long we must wait for the next event.
Core Concepts
The exponential distribution models the waiting time between consecutive events in a Poisson process. It is the continuous analog of the geometric distribution and the only continuous distribution that possesses the memoryless property.
Proof of the Memoryless Property
Interactive Visualization
CDF and Survival Function
Mean, Variance, and Higher Moments
MGF and Moments
Hazard Rate (Failure Rate)
Relationship to the Poisson Process
The Poisson-Exponential Connection
Worked Example
Gamma Distribution Connection
Specific Applications
- Queueing theory — Service times in M/M/1 queues are ; inter-arrival times are .
- Reliability engineering — Component lifetime before random failure (constant failure rate).
- Physics — Radioactive decay: time until a particle decays is where is the decay constant.
- Network traffic — Inter-packet arrival times in Poisson traffic models.