Mathematics.

probability distributions

Exponential Distribution

Statistics25 minDifficulty3 out of 10

You should know: poisson distribution, continuous probability distributions

Overview

The exponential distribution models the waiting time between independent events that occur at a constant average rate λ — it is the continuous counterpart to the Poisson distribution, which counts the number of such events in a fixed interval. It is characterized by a single rate parameter λ > 0, and is famous for its 'memoryless' property: the probability of waiting an additional t units of time is the same regardless of how long you've already waited. Common applications include modeling the time until the next customer arrival, the lifetime of electronic components, or the time between radioactive decay events.

Intuition

If events (like bus arrivals) happen randomly at a constant average rate, the amount of time you wait for the next one follows an exponential distribution. The memoryless property means that if you've already waited 10 minutes with no bus, your expected additional wait time is exactly the same as if you'd just started waiting — the process has no 'memory' of elapsed time, which is a direct consequence of events occurring independently at a constant rate (the same assumption underlying the Poisson distribution).

Formal Definition

Definition

If X is the waiting time until the next event in a Poisson process with rate λ, X ~ Exponential(λ), with:

f(x)=λe−λx,x≥0f(x) = \lambda e^{-\lambda x}, \qquad x \ge 0
Probability density function
F(x)=1−e−λx,x≥0F(x) = 1 - e^{-\lambda x}, \qquad x \ge 0
Cumulative distribution function
E[X]=1λ,Var⁡(X)=1λ2E[X] = \frac{1}{\lambda}, \qquad \operatorname{Var}(X) = \frac{1}{\lambda^2}
Mean and variance
P(X>s+t∣X>s)=P(X>t)P(X > s + t \mid X > s) = P(X > t)
Memoryless property

Worked Examples

  1. Apply the mean formula for the exponential distribution.

    E[X]=1λ=12=0.5E[X] = \frac{1}{\lambda} = \frac{1}{2} = 0.5

Answer: The expected wait is 0.5 minutes (30 seconds).

Practice Problems

Difficulty 3/10

For X ~ Exponential(λ = 4), find the mean and variance.

Difficulty 4/10

For X ~ Exponential(λ = 0.5), find P(X ≤ 2).

Difficulty 5/10

A component has an exponentially distributed lifetime with mean 10 years (λ = 0.1). Given that it has already lasted 5 years, what is the probability it lasts at least 5 more years?

Quiz

The exponential distribution is the continuous analogue of which discrete distribution?
The 'memoryless property' of the exponential distribution means:
For X ~ Exponential(λ), the mean is:

Summary

  • The exponential distribution models the waiting time between independent events occurring at a constant average rate λ.
  • Its mean is 1/λ and variance is 1/λ²; its CDF is F(x) = 1 - e^{-λx}.
  • It has the unique memoryless property: past elapsed waiting time gives no information about future waiting time.

References

  1. WebsiteWikipedia — Exponential distribution