# AP Statistics Curriculum 2007 Exponential

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+ | ==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Exponential Distribution== | ||

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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. | ||

- | <br />'''Probability density function''': For | + | <br />'''Probability density function''': For <math>X\sim \operatorname{Exponential}(\lambda)\!</math>, the exponential probability density function is given by |

:<math>\lambda e^{-\lambda x}\!</math> | :<math>\lambda e^{-\lambda x}\!</math> | ||

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where | where | ||

*e is the natural number (e = 2.71828…) | *e is the natural number (e = 2.71828…) | ||

- | *<math>\lambda</math> is the mean time between events | + | *<font size="3"><math>\lambda</math></font> is the mean time between events |

*x is a random variable | *x is a random variable | ||

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where | where | ||

*e is the natural number (e = 2.71828…) | *e is the natural number (e = 2.71828…) | ||

- | *<math>\lambda</math> is the mean time between events | + | *<font size="3"><math>\lambda</math></font> is the mean time between events |

*x is a random variable | *x is a random variable | ||

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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=== | ===Applications=== | ||

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*The height of various molecules in a gas at a fixed temperature and pressure in a uniform gravitational field | *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 | *The monthly and annual maximum values of daily rainfall and river discharge volumes | ||

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===Example=== | ===Example=== | ||

Suppose you usually get 2 phone calls per hour. Compute the probability that a phone call will arrive within the next hour. | 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: | + | 2 phone calls per hour means that we would expect one phone call every 1/2 hour so <font size="3"><math>\lambda=0.5</math></font>. 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> | :<math>P(0\le X\le 1)=\sum_{x=0}^1 0.5e^{-0.5x}=0.393469</math> | ||

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

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

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+ | <hr> | ||

+ | * SOCR Home page: http://www.socr.ucla.edu | ||

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+ | {{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Exponential}} |

## Current revision as of 22:31, 18 July 2011

## Contents |

## General Advance-Placement (AP) Statistics Curriculum - 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.

**Probability density function**: For , 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

- SOCR Home page: http://www.socr.ucla.edu

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