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The exponential distribution is a "waiting time" distribution. Suppose events occur independently and randomly through time with an average time between events of β. The time spent waiting for an event to occur is an exponential(β) random variable.

The exponential distribution is memoryless. This means that the probability of having to wait a further amount of time is independent of the time already spent waiting.

Parameter Range Description
β β > 0 Scale parameter

Probability Density Function

f ( x ; β ) = 1 β e − x / β
F ( x ; β ) = 1 − e − x / β
S ( x ; β ) = e − x / β
h ( x ; β ) = 1 β

Support

0 ≤ x < ∞

Mean

Variance

Example β
A radioactive substance emits two alpha particles every second on average. Let X be the waiting time for the next particle to be emitted. 0.5000
On a particular night, a shooting star occurs every five minutes on average. Let X be the waiting time for the next shooting star. 5.000
A hitchhiker sees one car every 10 minutes on average. Let X be the waiting time for the next car. 10.00

X ∼ Exponential(β)

β =
Chart of the exponential distribution Chart area for displaying the exponential pdf, cdf, survival, hazard, visualization, and simulation

E(X) = , Var(X) =

Note that the standard deviation equals the expected value. All exponential distributions have the same shape, and differ only in scale.

Note that the standard deviation equals the expected value. All exponential distributions have the same shape, and differ only in scale.

The graph above displays the survival function S(x) = P(X > x) = 1 - F(X), where F(x) is the cumulative distribution function (cdf).

Survival functions are used in survival analysis, a branch of statistics concerned with the expected duration until an event occurs such as death or the failure of a mechanical system.

The graph above displays the hazard function h(x). This equals f(x)/S(x), where f(x) is the pdf and S(x) = P(X > x) is the survival function.

Note that:

  • The hazard function is constant and independent of time, since the probability of an event occurring is independent of the time already spent waiting.
  • The hazard function is inversely proportional to the mean time β between events.

The illustration above shows events occuring randomly in time independently of other events, with a mean time between events of β. The time at which the first event occurs has an exponential(β) distribution.

The simulation above shows an event occuring randomly in time independently of other events, with a mean time between events of β. The light blue line shows the time at which the first event occurs, which has an exponential(β) distribution. The histogram accumulates the results of each simulation.

Y = e−X ∼ Beta(1/β, 1) Exponential(2) ∼ Chi-Squared(2) min Xᵢ ∼ Exponential(β/n) X₁/X₂ ∼ F(2, 2) ∑ Xᵢ ∼ Gamma(n, β) ⌈X⌉ ∼ Geometric(1 − e-1/β) limn→∞ max Xᵢ + μ − βlog n ∼ Gumbel(μ, β) -log X ∼ Gumbel(−log β, 1) X₁ − X₂ + μ ∼ Laplace(μ, β) eX ∼ Pareto(1/β, 1) 1 − e−X/β ∼ Standard Uniform X1/α ∼ Weibull(α, β1/α)

Chart of the related distribution Chart area for displaying the related pdf, cdf, and simulation

E(Y) = , Var(Y) =

Note that:

  • The support of X is [0, ∞), so the support of e−X is [0, 1].
  • e−X is a decreasing function. So, as the probability mass of X shifts to the right, the probability mass of Y shifts to the left (and vice-versa).

Note that a chi-squared(2) random variable is the sum of the squares of two independent standard normal random variables X₁ and X₂

The exponential(β) distribution is shown in red superimposed over the chi-squared(2). It can be seen that these are identical when β = 2.

This can be thought of as waiting for the first event to occur when there are n exponential(β) processes running in parallel. The average waiting time for each process is β, but the average waiting time until an event occurs in any of the processes is reduced to β/n.

The exponential(β) distribution is shown in red superimposed over the distribution of the sample minimum.

Note that the F distribution is a ratio distribution, for which the expected value and variance are undefined when ν₂ ≤ 2.

Since the exponential distribution models the time for one event to occur, the gamma distribution therefore models the time for n events to occur (when n is an integer).

Note that P(Y = 0) = 0, so the support of Y is 1, 2, 3, ...

This limit provides a motivation for calling the Gumbel an extreme value distribution.

The red dashed line shows the distribution of max{Xᵢ} + μ − βlog n, which can be seen to converge to a Gumbel(μ, β) distribution as n → ∞.

Note that changing the value of β in the exponential distribution just changes the location parameter of the Gumbel distribution while keeping the shape constant.

Note that the support of X₁ and X₂ is [0, ∞), so the support of X₁ − X₂ + μ is (−∞, ∞).

Note that:

  • The support of X is [0, ∞), so the support of eX is [1, ∞).
  • eX is an increasing function. So, as the probability mass of X shifts to the left or right, the probability mass of Y also shifts in the same direction.

This is an example of the probability integral transformation. The transformation can also be stated as Y = FX(X), where FX is the cumulative density function for the exponential(β) distribution. This type of transformation always results in a standard uniform random variable when FX is continuous.

Note that as α increases the probability mass becomes increasingly concentrated around the mean, while when α decreases the probability mass becomes more spread out.