AP Statistics Curriculum 2007 Exponential

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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

\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}
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