Difference between revisions of "Exponential distribution"
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| − | == Exponential( | + | == Exponential(r) == |
| − | Describes the distribution of times between successive independent | + | Describes the distribution of times between successive independent events in a Poisson process with an average rate of «r» events |
| − | events in a Poisson process with an average rate of | + | per unit time. The rate «r» is the reciprocal of the mean of the [[Poisson]] distribution—the average number of events per unit time. Its |
| − | per unit time. The rate | + | standard deviation is also ''1/r''. |
| − | distribution—the average number of events per unit time. Its | ||
| − | standard deviation is also 1/r. | ||
| − | A model with exponentially distributed times between events is | + | A model with exponentially distributed times between events is said to be Markov, implying that knowledge about when the next |
| − | said to be Markov, implying that knowledge about when the next | + | event occurs does not depend on the system's history or how much time has elapsed since the previous event. More general |
| − | event occurs does not depend on the system's history or how | + | distributions such as the gamma or Weibull do not exhibit this property. |
| − | much time has elapsed since the previous event. More general | ||
| − | distributions such as the gamma or Weibull do not exhibit this | ||
| − | property. | ||
== Density functions == | == Density functions == | ||
| + | The probability density of [[Exponential]](r) for <math>x \geq 0</math> is given by | ||
| + | :<code>p(x) = r*Exp(-r*x)</code> | ||
| − | The | + | The cumulative density function ([[CDF]]) is given by |
| − | + | :<code>F(x) = 1 - Exp(-r*x)</code> | |
| − | + | ==Optional parameters== | |
| − | + | === Over === | |
| − | + | :[[Exponential]](r, Over: I, J) | |
| − | == Over == | + | generates independent exponential distributions for each combination of elements in <code>I</code> and <code>J</code>. |
| − | |||
| − | |||
| − | |||
| − | generates independent exponential distributions for each combination of elements in I and J. | ||
== Parameter Estimation == | == Parameter Estimation == | ||
| − | + | Suppose you have sampled historic data in <code>X</code>, indexed by <code>I</code>, with each measurement being a positive real number, and you want to estimate the parameter for the best-fit exponential distribution. The parameter can be estimated using: | |
| − | Suppose you have sampled historic data in | + | :<code>1/Mean(X, I)</code> |
| − | :1 / | ||
== See Also == | == See Also == | ||
| − | + | * [[Exp]] | |
* [[Dens_Exponential]] | * [[Dens_Exponential]] | ||
* [[CumExponential]] | * [[CumExponential]] | ||
* [[CumExponentialInv]] | * [[CumExponentialInv]] | ||
* [[Gamma]] | * [[Gamma]] | ||
| + | * [[Mean]] | ||
Revision as of 23:40, 21 January 2016
Exponential(r)
Describes the distribution of times between successive independent events in a Poisson process with an average rate of «r» events per unit time. The rate «r» is the reciprocal of the mean of the Poisson distribution—the average number of events per unit time. Its standard deviation is also 1/r.
A model with exponentially distributed times between events is said to be Markov, implying that knowledge about when the next event occurs does not depend on the system's history or how much time has elapsed since the previous event. More general distributions such as the gamma or Weibull do not exhibit this property.
Density functions
The probability density of Exponential(r) for [math]\displaystyle{ x \geq 0 }[/math] is given by
p(x) = r*Exp(-r*x)
The cumulative density function (CDF) is given by
F(x) = 1 - Exp(-r*x)
Optional parameters
Over
- Exponential(r, Over: I, J)
generates independent exponential distributions for each combination of elements in I and J.
Parameter Estimation
Suppose you have sampled historic data in X, indexed by I, with each measurement being a positive real number, and you want to estimate the parameter for the best-fit exponential distribution. The parameter can be estimated using:
1/Mean(X, I)
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