The Exponential Distribution

The exponential distribution models the waiting time until the next event in a process where events occur at a constant average rate: the time between new infections, the length of an infectious period, the interval between radioactive decays, or the time until a component fails. It is the continuous companion of the Poisson distribution.

The exponential density; the mean 1/lambda marks the average waiting time.

Definition

Let $X\sim\mathrm{Exponential}(\lambda)$ be the waiting time with rate $\lambda$ . Its probability density function is $f(x)=\lambda e^{-\lambda x},\qquad x\ge 0,$ with cumulative distribution function $F(x)=1-e^{-\lambda x}$ .

Support: $x\in[0,\infty)$ .
Parameter: rate $\lambda>0$ (events per unit time). The reciprocal $1/\lambda$ is the mean waiting time, sometimes called the scale.
Mean: $\mathbb{E}[X]=\dfrac{1}{\lambda}$ .
Variance: $\mathrm{Var}(X)=\dfrac{1}{\lambda^2}$ .

The memoryless property

The exponential is the only continuous distribution that is memoryless: $P(X>s+t\mid X>s)=P(X>t).$ Having already waited $s$ units tells you nothing about how much longer you must wait — the process “forgets” the past. This is why it models the time to the next event in a constant-rate process, and why it is a common (if simplistic) model for infectious periods that end at a constant hazard.

Link to the Poisson

If events occur according to a Poisson process with rate $\lambda$ — so the count in any window is Poisson( $\lambda\cdot\text{length}$ ) — then the waiting times between consecutive events are independent Exponential( $\lambda$ ) variables. Counting events and timing events are two views of the same underlying process.

When it arises

The exponential arises for inter-event times at a constant rate: time between infections in early epidemic growth, duration of an infectious or latent period, survival times with constant hazard, and service or arrival times in queues.

In code

Watch the parametrization carefully. R uses the rate $\lambda$ , SciPy uses a scale $=1/\lambda$ , and Julia’s Exponential(θ) also uses the scale $\theta=1/\lambda$ .

R

# R parametrizes by rate = lambda
dexp(1, rate = 0.5)    # density at x = 1
pexp(1, rate = 0.5)    # P(X <= 1) = 1 - exp(-0.5)
qexp(0.95, rate = 0.5) # 95% quantile

set.seed(123)
x <- rexp(10000, rate = 0.5)  # random sample, mean = 1/0.5 = 2
hist(x, breaks = 40, freq = FALSE)     # histogram
curve(dexp(x, rate = 0.5), add = TRUE) # overlay the density

Python

import numpy as np
from scipy import stats

# scipy.stats.expon uses SCALE = 1/lambda (not the rate!)
lam = 0.5
stats.expon.pdf(1, scale=1/lam)    # density at 1
stats.expon.cdf(1, scale=1/lam)    # P(X <= 1)
stats.expon.ppf(0.95, scale=1/lam) # 95% quantile

rng = np.random.default_rng(123)
x = rng.exponential(scale=1/lam, size=10000)  # numpy also uses scale = 1/lambda
# plt.hist(x, bins=40, density=True); overlay stats.expon.pdf on a grid

Julia

using Distributions, Random

# Julia's Exponential(theta) uses the SCALE theta = 1/lambda, NOT the rate.
lam = 0.5
d = Exponential(1 / lam)  # scale theta = 1/0.5 = 2  => rate lambda = 0.5
pdf(d, 1)                 # density at 1
cdf(d, 1)                 # P(X <= 1)
quantile(d, 0.95)         # 95% quantile

Random.seed!(123)
x = rand(d, 10_000)       # random sample, mean = theta = 2
# histogram(x, normalize=:pdf); plot!(t -> pdf(d, t)) to overlay the density

Simulation

Many draws with rate $\lambda=0.5$ have empirical mean near $1/\lambda=2$ and variance near $1/\lambda^2=4$ .

set.seed(7)
x <- rexp(1e6, rate = 0.5)
mean(x)  # ~ 2.0  (theoretical mean = 1/lambda)
var(x)   # ~ 4.0  (theoretical variance = 1/lambda^2)

Why it matters for statistics

The exponential is the building block of survival analysis and reliability, the simplest hazard model, and the inter-arrival law of Poisson processes used throughout epidemiology and queueing theory. Understanding its rate-vs-scale parametrization prevents a common and costly software bug when moving between R, Python, and Julia.