Exponential Distribution

The exponential distribution models the waiting time until the next event in a memoryless, constant-rate process.

The density

The exponential distribution with rate λ has density f(x) = λ e^{−λx} for x ≥ 0. Its mean is 1/λ and its variance is 1/λ². It describes the time between events that occur independently at a steady average rate — the continuous partner of the Poisson count.

The memoryless property

Kronos motion — next scientists

The exponential is the only continuous distribution with no memory: P(X > s + t | X > s) = P(X > t). Having already waited s units tells you nothing about the additional wait. A component with exponential lifetime does not age; its failure rate is constant.

Link to Poisson

If events form a Poisson process with rate λ, the number in a fixed interval is Poisson(λt) and the gaps between consecutive events are Exponential(λ). The two distributions are two views of the same process — counts versus waiting times.

Sampling and use

python

import math, random
lam = 0.5
x = -math.log(1-random.random())/lam  # inverse-transform sample
print(round(x,3))

Exponential waiting times drive discrete-event simulation and queueing models, where arrivals and service completions are generated as exponential gaps. They are also the simplest lifetime model in reliability analysis, appropriate when failures are random rather than wear-driven.

When it fails

Real components often show increasing failure rates as they wear out, which the constant-rate exponential cannot capture. The more flexible Weibull distribution generalizes it to model aging or early-life failures.