Exponential distribution

In probability theory and statistics, the exponential distribution is a continuous probability distribution with the probability density function (pdf)

Probability density function of exponential distribution for λ = 0.5, 1.0, and 1.5.

f(x) = λe^{- λx}

for x ≥ 0 and where λ > 0 is a parameter of the distribution. Alternatively, the exponential distribution can be parameterized by the scale parameter 1/λ.

The cumulative distribution function is given by

$F(x) = 1-e^{-\lambda x} \,\!$ .

The inverse cumulative distribution function is

$F^{-1}(p) = \frac{-\ln(1-p)}{\lambda}$ .

An exponential(λ) random variable has the following properties:

first quartile: ln(4/3)/λ
median: ln(2)/λ
third quartile: ln(4)/λ
mean: μ = 1/λ
variance: σ² = 1/λ²
skewness: γ₁ = 2
kurtosis excess: γ₁ = 6
entropy: H = 1 − ln(λ) nats

The exponential distribution is used to model Poisson processes, which are situations in which an object initially in state A can change to state B with constant probability per unit time λ. The time at which the state actually changes is described by an exponential random variable with parameter λ. Therefore, the integral from 0 to T over f is the probability that the object is in state B at time T.

The exponential distribution may be viewed as a continuous counterpart of the geometric distribution, which describes the number of Bernoulli trials necessary for a discrete process to change state. In contrast, the exponential distribution describes the time for a continuous process to change state.

Examples of variables that are approximately exponentially distributed are:

the time until you have your next car accident
the time until you get your next phone call (assuming you get called many times a day, or get called by people from many different time zones)
the distance between mutations on a DNA strand
the distance between roadkill
the time until a radioactive particle decays
the number of dice rolls until you roll 6 11 times in a row

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

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

This says that the conditional probability that we need to wait, for example, more than another 10 seconds before the first arrival, given that the first arrival has not yet happened after 30 seconds, is no different from the initial probability that we need to wait more than 10 seconds for the first arrival. This is often misunderstood by students taking courses on probability: the fact that P(T > 40 | T > 30) = P(T > 10) does not mean that the events T > 40 and T > 10 are independent. To summarize: "memorylessness" of the probability distribution of the waiting time T until the first arrival means

$\mathrm{(Right)}\ P(T>40 \mid T>30)=P(T>10).$

It does not mean

$\mathrm{(Wrong)}\ P(T>40 \mid T>30)=P(T>40).$

(That would be independence. These two events are not independent.)

Generating variables with exponential distribution

Given a random variable Y with uniform distribution in the interval (0;1], the variable

$T=\frac{-\ln Y}{\lambda}$

has an exponential distribution with the parameter λ.

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Categories: Probability distributions

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