AP Statistics Curriculum 2007 Exponential

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(Exponential Distribution)
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'''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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<br />'''Probability density function''': For <math>X\sim Exponential(\lambda)\!</math>, the exponential probability density function is given by
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<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>

Revision as of 21:16, 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\sim \operatorname{Exponential}(\lambda)\!, the exponential probability density function is given by

\lambda e^{-\lambda x}\!

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

1-e^{-\lambda x}\!

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

M(t)=(1-\frac{t}{\lambda})^{-1}


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

E(X)=\frac{1}{\lambda}


Variance: The exponential variance is

Var(X)=\frac{1}{\lambda^2}

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:

P(0\le X\le 1)=\sum_{x=0}^1 0.5e^{-0.5x}=0.393469

The figure below shows this result using SOCR distributions

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