# AP Statistics Curriculum 2007 Exponential

### From Socr

### 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