In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
and
statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, the exponential distribution is the
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
of the time between events in a
Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the
gamma distribution
In probability theory and statistics, the gamma distribution is a two- parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma di ...
. It is the continuous analogue of the
geometric distribution, and it has the key property of being
memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.
The exponential distribution is not the same as the class of
exponential families of distributions. This is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes many other distributions, like the
normal,
binomial
Binomial may refer to:
In mathematics
*Binomial (polynomial), a polynomial with two terms
*Binomial coefficient, numbers appearing in the expansions of powers of binomials
*Binomial QMF, a perfect-reconstruction orthogonal wavelet decomposition
* ...
,
gamma
Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter r ...
, and
Poisson distributions.
Definitions
Probability density function
The
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
(pdf) of an exponential distribution is
:
Here ''λ'' > 0 is the parameter of the distribution, often called the ''rate parameter''. The distribution is supported on the interval . If a
random variable ''X'' has this distribution, we write .
The exponential distribution exhibits
infinite divisibility.
Cumulative distribution function
The
cumulative distribution function is given by
:
Alternative parametrization
The exponential distribution is sometimes parametrized in terms of the
scale parameter , which is also the mean:
Properties
Mean, variance, moments, and median
The mean or
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
of an exponentially distributed random variable ''X'' with rate parameter ''λ'' is given by
In light of the examples given
below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
, 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
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
of ''X'' is given by
so the
standard deviation is equal to the mean.
The
moments of ''X'', for
are given by
The
central moments of ''X'', for
are given by
where !''n'' is the
subfactorial of ''n''
The
median of ''X'' is given by
where refers to the
natural logarithm. Thus the
absolute difference between the mean and median is
in accordance with the
median-mean inequality.
Memorylessness
An exponentially distributed random variable ''T'' obeys the relation
This can be seen by considering the
complementary cumulative distribution function:
When ''T'' is interpreted as the waiting time for an event to occur relative to some initial time, this relation implies that, if ''T'' is conditioned on a failure to observe the event over some initial period of time ''s'', the distribution of the remaining waiting time is the same as the original unconditional distribution. For example, if an event has not occurred after 30 seconds, the
conditional probability that occurrence will take at least 10 more seconds is equal to the unconditional probability of observing the event more than 10 seconds after the initial time.
The exponential distribution and the
geometric distribution are
the only memoryless probability distributions.
The exponential distribution is consequently also necessarily the only continuous probability distribution that has a constant
failure rate
Failure rate is the frequency with which an engineered system or component fails, expressed in failures per unit of time. It is usually denoted by the Greek letter λ (lambda) and is often used in reliability engineering.
The failure rate of a ...
.
Quantiles
The
quantile function (inverse cumulative distribution function) for Exp(''λ'') is
The
quartiles are therefore:
*first quartile: ln(4/3)/''λ''
*
median: ln(2)/''λ''
*third quartile: ln(4)/''λ''
And as a consequence the
interquartile range
In descriptive statistics, the interquartile range (IQR) is a measure of statistical dispersion, which is the spread of the data. The IQR may also be called the midspread, middle 50%, fourth spread, or H‑spread. It is defined as the differen ...
is ln(3)/''λ''.
Kullback–Leibler divergence
The directed
Kullback–Leibler divergence in
nats of
("approximating" distribution) from
('true' distribution) is given by
Maximum entropy distribution
Among all continuous probability distributions with
support
Support may refer to:
Arts, entertainment, and media
* Supporting character
Business and finance
* Support (technical analysis)
* Child support
* Customer support
* Income Support
Construction
* Support (structure), or lateral support, a ...
and mean ''μ'', the exponential distribution with ''λ'' = 1/''μ'' has the largest
differential entropy. In other words, it is the
maximum entropy probability distribution
In statistics and information theory, a maximum entropy probability distribution has entropy that is at least as great as that of all other members of a specified class of probability distributions. According to the principle of maximum entrop ...
for a
random variate ''X'' which is greater than or equal to zero and for which E
'X''is fixed.
Distribution of the minimum of exponential random variables
Let ''X''
1, …, ''X''
''n'' be
independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independe ...
exponentially distributed random variables with rate parameters ''λ''
1, …, ''λ
n''. Then
is also exponentially distributed, with parameter
This can be seen by considering the
complementary cumulative distribution function:
The index of the variable which achieves the minimum is distributed according to the categorical distribution
A proof can be seen by letting
. Then,
Note that
is not exponentially distributed, if ''X''
1, …, ''X''
''n'' do not all have parameter 0.
Joint moments of i.i.d. exponential order statistics
Let
be
independent and identically distributed exponential random variables with rate parameter ''λ''.
Let
denote the corresponding
order statistics.
For
, the joint moment
of the order statistics
and
is given by
This can be seen by invoking the
law of total expectation and the memoryless property:
The first equation follows from the
law of total expectation.
The second equation exploits the fact that once we condition on
, it must follow that
. The third equation relies on the memoryless property to replace