Exponential distribution
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In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. They are often used to model the time between events that happen at a constant average rate.
Specification of the exponential distribution
Probability density function
The probability density function (pdf) of an exponential distribution has the form
- [f(x;\lambda) = \left\\lambda e^ &,\; x \ge 0, \\0 &,\; x < 0.\end\right.]
The exponential distributions can alternatively be parameterized by a scale parameter μ = 1/λ.
Cumulative distribution function
The cumulative distribution function is given by
- [F(x;\lambda) = \left\1-e^&,\; x \ge 0, \\0 &,\; x < 0.\end\right.]
Alternate specification
A commonly used alternate specification is to define the probability density function (pdf) of an exponential distribution as
- [f(x;\lambda) = \left\\frac e^ &,\; x \ge 0, \\0 &,\; x < 0.\end\right.]
This alternate specification is sometimes more convenient than the one given above, and some authors will use it as a standard definition. We shall not assume this alternate specification. Unfortunately this gives rise to a notational ambiguity. In general, the reader must check which of these two specifications is being used if an author writes "X ~ Exponential(λ)."
Occurrence and applications
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.
In real-world scenarios, the assumption of a constant rate (or probability per unit time) is rarely satisfied. For example, the rate of incoming phone calls differs according to the time of day. But if we focus on a time interval during which the rate is roughly constant, such as from 2 to 4 p.m. during work days, the exponential distribution can be used as a good approximate model for the time until the next phone call arrives. Similar caveats apply to the following examples which yield approximately exponentially distributed variables:
- the time until you have your next car accident;
- the time until a radioactive particle decays, or the time between beeps of a geiger counter;
- the number of dice rolls needed until you roll a six 11 times in a row;
- the time until a large meteor strike causes a mass extinction event.
Reliability theory and reliability engineering also make extensive use of the exponential distribution. Because of the memoryless property of this distribution, it is well-suited to model the constant hazard rate portion of the bathtub curve used in reliability theory. It is also very convenient because it is so easy to add failure rates in a reliability model. The exponential distribution is however not appropriate to model the overall lifetime of organisms or technical devices, because the "failure rates" here are not constant: more failures occur for very young and for very old systems.
In physics, if you observe a gas at a fixed temperature and pressure in a uniform gravitational field, the heights of the various molecules also follow an approximate exponential distribution. This is a consequence of the entropy property mentioned below.
Properties
Mean and variance
The mean or expected value of an exponentially distributed random variable X with rate parameter λ is given by
- [\mathbf[X] = \frac]
The variance of X is given by
- [\mathbf[X] = \frac].
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
- [P(T > s + t\; |\; T > t) = P(T > s) \;\; \hbox\ s, t \ge 0. ]
- [\mathrm\ P(T>40 \mid T>30)=P(T>10).]
- [\mathrm\ P(T>40 \mid T>30)=P(T>40).]
The exponential distributions and the geometric distributions are the only memoryless probability distributions.
The exponential distribution also has a constant hazard function.
Quartiles
The quantile function (inverse cumulative distribution function) for Exponential(λ) is
- [F^(p;\lambda) = \frac, \!]
- first quartile
- [\ln(4/3)/\lambda\,]
- median
- [\ln(2)/\lambda\,]
- third quartile
- [\ln(4)/\lambda\,]
Entropy
Among all continuous probability distributions with supportParameter estimation
Suppose you know that a given variable is exponentially distributed and you want to estimate the rate parameter λ.Maximum likelihood
The likelihood function for λ, given an independent and identically distributed sample x = (x1, ..., xn) drawn from your variable, is
- [ L(\lambda) = \prod_^n \lambda \, \exp(-\lambda x_i) = \lambda^n \, \exp\!\left(\!-\lambda \sum_^n x_i\right)=\lambda^n\exp\left(-\lambda n \overline\right) ]
- [\overline=\sum_^n x_i]
The derivative of the likelihood function's logarithm is
- [\frac}\lambda} \ln L(\lambda) = \frac}\lambda} \left( n \ln(\lambda) - \lambda n\overline \right) = -n\overline\ \left\ > 0 & \mbox\ 0 < \lambda < 1/\overline, \\ \\ = 0 & \mbox\ \lambda = 1/\overline, \\ \\ < 0 & \mbox\ \lambda > 1/\overline. \end\right. ]
- [\widehat = \frac1}].
Bayesian inference
The conjugate prior for the exponential distribution is the gamma distribution (of which the exponential distribution is a special case). The following parameterization of the gamma pdf is useful:
- [ \mathrm(\lambda \,;\, \alpha, \beta) = \frac} \, \lambda^ \, \exp(-\lambda\,\beta). \!]
- :[p(\lambda) \propto L(\lambda) \times \mathrm(\lambda \,;\, \alpha, \beta)]
- [= \lambda^n \, \exp(-\lambda\,n\overline) \times \frac} \, \lambda^ \, \exp(-\lambda\,\beta)]
- [\propto \lambda^ \, \exp(-\lambda\,(\beta + n\overline)).]
- [ p(\lambda) = \mathrm(\lambda \,;\, \alpha + n, \beta + n \overline). ]
Generating exponential variates
A conceptually very simple method for generating exponential variates is based on inverse transform sampling: Given a random variate U drawn from the uniform distribution on the unit interval [(0; 1)], the variate
- [T = F^(U) \!]
- [F^(p)=\frac. \!]
- [T = \frac. \!]
Related distributions
- An exponential distribution is a special case of a gamma distribution if [\alpha = 1] (or [k=1] depending on the parameter set used).
- [Y \sim \mathrm(\gamma, \lambda)] is a Weibull distribution if [Y = X^\,] and [X \sim \mathrm(\lambda^)]. In particular, every exponential distribution is also a Weibull distribution.
- [Y \sim \mathrm(1/\lambda)] is a Rayleigh distribution if [Y = \sqrt] and [X \sim \mathrm(\lambda)].
- [Y \sim \mathrm(\mu, \beta)] is a Gumbel distribution if [Y = \mu - \beta \log(X/\lambda)\,] and [X \sim \mathrm(\lambda)].
- [Y \sim \mathrm] is a Laplace distribution if [Y = X_1 - X_2] for two independent exponential distributions [X_1] and [X_2].
- [Y \sim \mathrm] is an exponential distribution if [Y = \min(X_1, X_2, \cdots, X_N)] for independent exponential distributions [X_i].
- [Y \sim \mathrm] is a gamma distribution if [Y = \sum_ X_i\,] for independent exponential distributions [X_i\,].
- [Y \sim \mathrm(0,1)] is a uniform distribution if [Y = \exp(-X/\lambda)\,] and [X \sim \mathrm(\lambda)].
- [X \sim \chi_2^2] is a chi-square distribution (with 2 degrees of freedom) if [X \sim \mathrm(\lambda = 2)].
References
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