Exponential distribution

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Exponential distributions describe the times between events in a Poisson process, i.e. a process in which events occur continuously and independently at a constant average rate.


Contents

[edit] Characterization

[edit] Quick Reference

Distribution Name (memorize) expresses ... (memorize)
Exponential with rate or inverse scale λ the times between events in a Poisson process.
Exponential mean \frac{1}{\lambda}
Exponential variance \frac{1}{\lambda^2}
Exponential (pdf) f(x)=\lambda e^{-\lambda x}, x \geq 0
Exponential (cdf) P(X < x) = P(X < x) = 1 − e − λx
Exponential MGF M(t) = \left( 1-\frac t\lambda \right)^{-1}

[edit] Probability density function

The probability density function (pdf) of an exponential distribution is

f(x;\lambda) = \left\{\begin{matrix} \lambda e^{-\lambda x}, &\; x \ge 0, \\ 0, &\; x < 0. \end{matrix}\right.

Here λ > 0 is the parameter of the distribution, often called the rate parameter. The distribution is supported on the interval [0, ∞). If a random variable X has this distribution, we write X ~ Exp(1/λ).

[edit] Cumulative distribution function

The cumulative distribution function is given by

F(x;\lambda) = P(X<x) = \left\{\begin{matrix} 1-e^{-\lambda x}, &\; x \ge 0, \\ 0, &\; x < 0. \end{matrix}\right.

[edit] Properties

[edit] Mean and variance

The mean or expected value of an exponentially distributed random variable X with rate parameter λ is given by

\mathrm{E}[X] = \frac{1}{\lambda}. \!

In light of the examples given above, this makes sense: if you receive phone calls at an average rate of 2 per hour, then you can expect to wait half an hour for every call.

The variance of X is given by

\mathrm{Var}[X] = \frac{1}{\lambda^2}. \!

[edit] Memorylessness

An important property of the exponential distribution is that it is memoryless. This means that if a random variable T is exponentially distributed, its conditional probability obeys

\Pr(T > s + t\; |\; T > s) = \Pr(T > t) \;\; \hbox{for all}\ s, t \ge 0.

The exponential distributions and the geometric distributions are the only memoryless probability distributions.

[edit] Related distributions


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