# 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

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