# AP Statistics Curriculum 2007 Exponential

## Contents

## General Advance-Placement (AP) Statistics Curriculum - Exponential Distribution

### Exponential Distribution

**Definition**: Exponential distribution, also known as Mean-Time-To-Failure 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…)
- \(\lambda\) 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…)
- \(\lambda\) 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 \(\lambda=0.5\). 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

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

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