Exponential distribution
The exponential distribution ( negative exponential distribution ) is a continuous probability distribution over the set of positive real numbers which is given by an exponential function. It is used as a model primarily in answering the question of the length of random time intervals, such as
- Time between two calls
- Lifetime of atoms during radioactive decay
- Life of components, machinery and equipment, if aging does not have to be considered.
- As a rough model for small and medium damage to household goods, motor vehicle liability, collision in the actuarial
Stands for the number of expected events per unit interval. As can be seen from the diagram, shorter intervals between events (interval length) are more likely. More rarely occur but also have very long intervals. The probability density may well values > 1 assume (for example ), since the area under the curve is normalized to 1. Specific information on the occurrence probability of the next event to win here most of the distribution function.
Often, the actual distribution is not exponential, but the exponential distribution is easy to handle and it is assumed for simplicity. It is applicable when a Poisson process is present, ie the Poisson assumptions are met.
- 3.1 Relationship to the continuous uniform distribution
- 3.2 Relationship with the normal distribution
- 3.3 Relationship to the geometric distribution
- 3.4 Relationship to the gamma distribution
- 3.5 Relationship to gamma-gamma distribution
- 3.6 Relationship to the Pareto distribution
- 3.7 Relationship to the Poisson distribution
- 3.8 Relationship to the Erlang distribution
- 3.9 Relationship to the Weibull distribution
- 3:10 relations with the χ ² - distribution
Definition
A continuous random variable satisfies the exponential distribution with the positive real parameters when the density function
Possesses.
The parameter has the character of an event rate and the distance of an event ( mean range or average life ).
A (especially in Anglo-Saxon common ) alternative parameterization leads to the probability density
The relationship with the above parameterization is simple. To avoid misunderstanding, it is recommended that the expectation value explicitly specify, so to speak of an exponential distribution with expected value.
Properties
Distribution function
The (cumulative ) distribution function of the exponential distribution
It allows the calculation of the probability of the occurrence of the next event in the interval from 0 to x.
The probability of the interval length is greater than x to the next event.
Expected value
The exponential distribution has the expected value, because
, The expected value corresponds to the average operating time of components, machinery and equipment, if aging do not have to be considered. He is referred to in this context as Mean Time Between Failures ( MTBF).
Median
The exponential distribution has its median at
Mode
The maximum value of the density function of the exponential case, i.e., the mode is
Variance
The variance is obtained analogously by
Standard deviation
For the standard deviation results
Coefficient of variation
From the expected value and variance are obtained directly the coefficient of variation. It is
So true
Other measures of variation
The mean absolute deviation
Is smaller than the standard deviation, the mean absolute deviation of the median with respect to
Is even smaller.
Skew
The skewness has independently of the parameter always has the value 2
Curvature
The curvature possesses independently of the parameter always has the value 9
Characteristic function
The characteristic feature is in the form
Moment generating function
The moment generating function of the exponential distribution is
Entropy
The entropy is the exponential
Probability of survival
Since the exponential distribution is used as a lifetime distribution and in technical areas as an expression for the reliability of a device, it is possible related variables such as survival function and the failure rate should be indicated using the distribution function. So it is called the complement of the distribution function of the survival function:
This results immediately, as to a time -related survival probability
The exponential distribution is memoryless lifetime distribution, ie the probability of survival with respect to a given time is independent of the previously achieved age. In contrast to the Weibull distribution, the exponential distribution can be used for so-called fatigue- free systems
The failure rate is given by
It is temporally and spatially constant for the exponential distribution and is commonly referred to in the literature with the constant λ.
Memoryless
The exponential distribution is in the following sense " memoryless ": it is known that an exponentially distributed random variable exceeds the value, then the conditional probability that it exceeds at least as large as the that a exponentially distributed random variables ( with the same parameters ) exceeds the value, formal
This behavior is also called Markov property.
The memoryless is even a defining property of the exponential distribution; this is the only possible continuous distribution with this property. This follows directly by the definition of conditional probability and the Cauchy functional equation. The discrete counterpart to this is the geometric distribution as the only possible discrete memoryless distribution.
Other properties
Are stochastically independent, then
Relationship to other distributions
Relationship to the continuous uniform distribution
If a uniformly distributed on the interval [ 0, 1] continuous random variable, then the exponential distribution with parameter is sufficient.
Relation to the normal distribution
Are the random variables and standard normally distributed and independent, then is exponentially distributed with parameter.
Relationship to the geometric distribution
In analogy to the discrete geometric distribution, the continuous exponential distribution determines the waiting time to the first occurrence of an event that occurs according to a Poisson process; the geometric distribution can therefore be regarded as a discrete equivalent of the exponential distribution.
Relationship to the gamma distribution
- The generalization of the exponential distribution, ie the waiting time until the arrival of the -th event of a Poisson process is described by the gamma distribution. The exponential distribution with parameter is identical to the gamma distribution with parameters and. The exponential therefore also has all the properties of the gamma distribution. In particular, the sum of independent - distributed random variables is gamma or Erlangverteilt with parameters and.
- The convolution of two exponential distributions with the same results in a gamma distribution with.
Relation to the gamma distribution gamma
If the parameter of the exponential distribution is a random variable that is distributed as a gamma distribution, then the company resulting random variable as a gamma-gamma distribution is distributed.
Relationship with the Pareto distribution
If Pareto distributed with parameters and, then is exponentially distributed with parameter.
Relationship to the Poisson distribution
The distances between the occurrence of random events can often be described by the exponential distribution. In particular, the distance between two consecutive events of a Poisson process with rate exponentially distributed with the parameter. In this case, the number of events in an interval of the length is a Poisson distribution with parameter.
Derivation: Let w be a local or a time variable and little constant occurrence frequency of events in the unit interval of w. Then one finds the Poisson assumptions, the probability for the next occurrence of an event in the small interval as the product of the probability to have no event to w and one in the interval:
It follows after division by the probability density of the exponential distribution with an event rate than middle- and event distance.
Relationship with the Erlang distribution
- For a Poisson process the random number of events up to a defined point in time using Poisson distribution is determined, the random time until th event is Erlang distributed. In case this is Erlang distribution over into an exponential distribution with which the time to the first random event and the time can be determined between two consecutive events.
- The sum of independent exponentially distributed random variables has the Erlang distribution of order.
Relationship with the Weibull distribution
- With the Weibull distribution merges with the exponential distribution. In other words, the exponential distribution deals with problems with constant failure rate. If we examine, however, problems with rising or falling failure rate, then you go from the exponential distribution for Weibull distribution about.
- When exponentially - distributed, then Weibull distribution is.
Relations with the χ ² - distribution
The χ ² - distribution goes in for the exponential distribution with the parameter.
Example of use
The exponential distribution is a typical lifetime distribution. So the life of electronic components, for example, often almost exponentially. Here, especially the memoryless plays a significant role: the probability that an x days old device still holds at least t days, is therefore the same as the that a new device holds at all t days. Characteristic of the exponential distribution is the constant failure rate.
This is only approximately correct for example incandescent lamps because they are heavily used only on power. In living beings no exponential distribution may also be used, otherwise the example would be the probability that an octogenarian another fifty years lives, just as high as the that a newborn reaches the fiftieth year.
Example: In an electronics company radio alarm clock are produced. As part of the quality assurance examines the duration of the alarm clock function on the basis of complaints. It turns out that on average fail regardless of their age per day 5 ‰ of the alarm clock.
The random variable " duration of the functioning of a radio alarm clock in days " is therefore exponentially distributed with failure rate. Correspondingly, the average time until a clock fails, days.
The probability that more than one alarm (yet) holds 20 days,
That is, after 20 days are on average about 10 % of the alarm clock failed.
Accordingly, the proportion of alarm clock that can withstand at least 180 days,
So keep an average of about 40 % of the alarm clock more than 180 days.
Although the exponential lifetime distribution at the beginning in absolute terms more devices fail, the failure rate is constant: in each time interval fall in relative terms is always the same number of devices. This fact should not be confused with the early failure of the bathtub curve. Here is the beginning, the failure rate is higher and is not constant over the lifetime. Description of the bathtub curve to another lifetime distribution ( Weibull distribution ) is required.
Random numbers
To generate exponentially distributed random numbers to offer the inversion method.
The to be formed after the Simulationslemma inverse of the cumulative distribution function is here. To a series of standard random numbers can therefore be a consequence of exponentially distributed random numbers to calculate. One can instead also be expected.