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Exponential distribution

Lecture




Indicative distribution
Probability density
  Exponential distribution
Distribution function
  Exponential distribution
Designation   Exponential distribution
Options   Exponential distribution - intensity or inverse scale factor
Carrier   Exponential distribution
Probability density   Exponential distribution
Distribution function   Exponential distribution
Expected value   Exponential distribution
Median   Exponential distribution
Fashion   Exponential distribution
Dispersion   Exponential distribution
Asymmetry coefficient   Exponential distribution
Coefficient of kurtosis   Exponential distribution
Informational entropy   Exponential distribution
Generating function of moments   Exponential distribution
Characteristic function   Exponential distribution

The exponential or exponential distribution is an absolutely continuous distribution that simulates the time between two successive accomplishments of the same event.

Content

  • 1 Definition
  • 2 distribution function
  • 3 Moments
  • 4 lack of memory
  • 5 Relationship with other distributions

Definition [edit]

Random value   Exponential distribution has an exponential distribution with the parameter   Exponential distribution if its density is

  Exponential distribution .

Example. Suppose there is a store in which buyers enter from time to time. Under certain assumptions, the time between occurrences of two consecutive buyers will be a random variable with an exponential distribution. The average waiting time for a new customer (see below) is   Exponential distribution . Parameter itself   Exponential distribution then it can be interpreted as the average number of new customers per unit of time.

In this paper, for definiteness, we will assume that the density of an exponential random variable   Exponential distribution given by the first equation, and we will write:   Exponential distribution .

Distribution function [edit]

Integrating the density, we obtain the exponential distribution function:

  Exponential distribution

Moments [edit]

By simple integration, we find that the generating function of moments for the exponential distribution has the form:

  Exponential distribution ,

where do we get all the moments:

  Exponential distribution .

In particular,

  Exponential distribution ,
  Exponential distribution ,
  Exponential distribution .

Lack of memory [edit]

Let be   Exponential distribution . Then   Exponential distribution .

Example. Let the buses come to a stop randomly, but with some fixed average intensity. Then the amount of time already spent by the passenger on waiting for the bus does not affect the time that he still has to wait.

Relationship with other distributions [edit]

  • A minimum of independent exponential random variables is also an exponential random variable. Let be   Exponential distribution independent random variables, and   Exponential distribution . Then
  Exponential distribution .
  • Exponential distribution is a special case of Gamma distribution:
  Exponential distribution .
  • The sum of independent identically distributed exponential random variables has a gamma distribution. Let be   Exponential distribution independent random variables, and   Exponential distribution . Then
  Exponential distribution .
  • The exponential distribution can be obtained from a continuous uniform distribution using the inverse transform method. Let be   Exponential distribution . Then
  Exponential distribution .
  • Exponential distribution with parameter   Exponential distribution - This is a special case of the chi-square distribution:
  Exponential distribution
  • The exponential distribution is a special case of the Weibull distribution.

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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis