# AP Statistics Curriculum 2007 Exponential

### From Socr

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- | === Exponential Distribution === | + | ===Exponential Distribution=== |

'''Definition''': Exponential distribution is a special case of the gamma distribution. Whereas the gamma distribution is the waiting time for more than one event, the exponential distribution describes the time between a single Poisson event. | '''Definition''': Exponential distribution is a special case of the gamma distribution. Whereas the gamma distribution is the waiting time for more than one event, the exponential distribution describes the time between a single Poisson event. | ||

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:<math>Var(X)=\frac{1}{\lambda^2}</math> | :<math>Var(X)=\frac{1}{\lambda^2}</math> | ||

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+ | ===Applications=== | ||

+ | The exponential distribution occurs naturally when describing the waiting time in a homogeneous Poisson process. It can be used in a range of disciplines including queuing theory, physics, reliability theory, and hydrology. Examples of events that may be modeled by exponential distribution include: | ||

+ | *The time until a radioactive particle decays | ||

+ | *The time between clicks of a Geiger counter | ||

+ | *The time until default on payment to company debt holders | ||

+ | *The distance between roadkills on a given road | ||

+ | *The distance between mutations on a DNA strand | ||

+ | *The time it takes for a bank teller to serve a customer | ||

+ | *The height of various molecules in a gas at a fixed temperature and pressure in a uniform gravitational field | ||

+ | *The monthly and annual maximum values of daily rainfall and river discharge volumes | ||

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+ | |||

+ | ===Example=== | ||

+ | Suppose you usually get 2 phone calls per hour. Compute the probability that a phone call will arrive within the next hour. | ||

+ | |||

+ | 2 phone calls per hour means that we would expect one phone call every 1/2 hour so <math>\lambda=0.5</math>. We can then compute this as follows: | ||

+ | |||

+ | :<math>P(0\le X\le 1)=\sum_{x=0}^1 0.5e^{-0.5x}=0.393469</math> | ||

+ | |||

+ | The figure below shows this result using [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html SOCR distributions] | ||

+ | <center>[[Image:Exponential.jpg|600px]]</center> |

## Revision as of 18:46, 11 July 2011

### Exponential Distribution

**Definition**: Exponential distribution is a special case of the gamma distribution. Whereas the gamma distribution is the waiting time for more than one event, the exponential distribution describes the time between a single Poisson event.

**Probability density function**: For X~Exponential(λ), the exponential probability density function is given by

where

- e is the natural number (e = 2.71828…)
- λ is the mean time between events
- x is a random variable

**Cumulative density function**: The exponential cumulative distribution function is given by

where

- e is the natural number (e = 2.71828…)
- λ is the mean time between events
- x is a random variable

**Moment generating function**: The exponential moment-generating function is

**Expectation**: The expected value of a exponential distributed random variable x is

**Variance**: The exponential variance is

### Applications

The exponential distribution occurs naturally when describing the waiting time in a homogeneous Poisson process. It can be used in a range of disciplines including queuing theory, physics, reliability theory, and hydrology. Examples of events that may be modeled by exponential distribution include:

- The time until a radioactive particle decays
- The time between clicks of a Geiger counter
- The time until default on payment to company debt holders
- The distance between roadkills on a given road
- The distance between mutations on a DNA strand
- The time it takes for a bank teller to serve a customer
- The height of various molecules in a gas at a fixed temperature and pressure in a uniform gravitational field
- The monthly and annual maximum values of daily rainfall and river discharge volumes

### Example

Suppose you usually get 2 phone calls per hour. Compute the probability that a phone call will arrive within the next hour.

2 phone calls per hour means that we would expect one phone call every 1/2 hour so λ = 0.5. We can then compute this as follows:

The figure below shows this result using SOCR distributions