Exponential Distribution: Waiting Time Until an Event Occurs

The exponential distribution models an experiment where events occur continuously and independently at a constant average rate, and the random variable measures the waiting time until the next event occurs. There are no discrete trials and no fixed endpoint; time flows continuously until the event happens. The defining feature of this experiment is that the probability of waiting longer depends only on the current moment, not on how much time has already passed.

The Probabilistic Experiment Behind exponential distribution

The probabilistic experiment behind the exponential distribution focuses on waiting time rather than counts or measurements. The experiment observes how long it takes until a particular event happens, under the assumption that the event occurs continuously and unpredictably, but at a steady average rate.

A defining characteristic of this experiment is lack of memory. The process does not “age”: the chance that the event occurs in the next instant does not depend on how much time has already passed. Waiting longer provides no advantage or disadvantage—the system resets at every moment.

This framework applies when events occur independently, one at a time, and without buildup or anticipation. Time is continuous, and probability accumulates smoothly as time passes.

The exponential distribution captures the idea that short waiting times are common, while long waits are possible but become progressively more rare.

Example:

Consider the time until the next phone call arrives at a quiet call center where calls come in at a stable average rate. Whether the last call arrived a second ago or an hour ago does not affect the likelihood of receiving the next one in the coming minute.

See More Examples of Exponential →

Notation

X \sim \text{Exp}(\lambda)

— distribution of the random variable (rate parameterization).

X \sim \text{Exponential}(\lambda)

— alternative explicit form.

\text{Exp}(\lambda)

— used to denote the distribution itself (not the random variable).

X \sim \text{Exp}(\beta)

— scale parameterization where

\beta = 1/\lambda

is the mean.

Note: Always check whether the parameter represents the rate (λ) or the scale (β = 1/λ). Statistical software may use either convention. The rate parameterization is more common in probability theory.

See All Probability Symbols and Notations →

Parameters

λ (lambda): rate parameter, where

\lambda > 0

The exponential distribution is fully characterized by a single parameter.
λ represents the average rate at which events occur per unit time.
Alternatively, the distribution is sometimes parameterized by

\beta = 1/\lambda

, the scale parameter, which represents the mean waiting time.
Higher λ means events occur more frequently (shorter waiting times), while lower λ means events are more rare (longer waiting times).

Probability Density Function (PDF) and Support (Range)

Exponential Distribution

Time between events in a Poisson process

Rate Parameter (λ): 1.00

Explanation

The exponential distribution models the time between events in a Poisson process, where events occur continuously and independently at a constant average rate. The probability density function is $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$ , where $\lambda$ is the rate parameter. The expected value is $E[X] = \frac{1}{\lambda}$ and the variance is $\text{Var}(X) = \frac{1}{\lambda^2}$ . The exponential distribution has the memoryless property, meaning the probability of an event occurring in the next interval is independent of how much time has already passed. Common applications include time until equipment failure, waiting time between customer arrivals, radioactive decay, and time until the next earthquake.

Cumulative Distribution Function (CDF)

Exponential Distribution CDF

Visualizing probability accumulation for exponential distribution

Exponential - CDF

Rapid initial rise, asymptotic approach to 1

Rate Parameter (λ): 1.00

CDF Explanation

The cumulative distribution function (CDF) of the exponential distribution is

F(x) = 1 - e^{-\lambda x}

for

x \geq 0

, and

F(x) = 0

for

x < 0

. The CDF shows the probability that the waiting time is less than or equal to

x

. The curve rises quickly at first and then asymptotically approaches 1. The memoryless property of the exponential distribution means that

P(X > s + t | X > s) = P(X > t)

for all

s, t \geq 0

. The median waiting time (where

F(x) = 0.5

) is

\frac{\ln(2)}{\lambda}

, and about 63.2% of events occur within the first

\frac{1}{\lambda}

time units.

Exponential Distribution

Exponential Distribution: Waiting Time Until an Event Occurs

The Probabilistic Experiment Behind exponential distribution

Notation

Parameters

Probability Density Function (PDF) and Support (Range)

Exponential Distribution

Explanation

Cumulative Distribution Function (CDF)

Exponential Distribution CDF

Exponential - CDF

CDF Explanation

Expected Value (mean)

Variance and Standard Deviation

Mode and Median

Quantiles/Percentiles

Moment Generating Function

Real-World Examples and Common Applications

Interactive Calculator

Special Cases

Properties

Related Distributions

Parameter Estimation