Exponential distribution

Lecture

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

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 has an exponential distribution with the parameter if its density is

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 . Parameter itself 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: .

Distribution function [edit]

Integrating the density, we obtain the exponential distribution function:

Moments [edit]

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

where do we get all the moments:

In particular,

Lack of memory [edit]

Let be Exponential distribution . Then .

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.