AP Statistics Curriculum 2007 Exponential

From SOCR
Revision as of 18:39, 18 July 2011 by JayZzz (talk | contribs)
Jump to: navigation, search

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

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…)
  • \(\lambda\) 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

Exponential.jpg





Translate this page:

(default)
Uk flag.gif

Deutsch
De flag.gif

Español
Es flag.gif

Français
Fr flag.gif

Italiano
It flag.gif

Português
Pt flag.gif

日本語
Jp flag.gif

България
Bg flag.gif

الامارات العربية المتحدة
Ae flag.gif

Suomi
Fi flag.gif

इस भाषा में
In flag.gif

Norge
No flag.png

한국어
Kr flag.gif

中文
Cn flag.gif

繁体中文
Cn flag.gif

Русский
Ru flag.gif

Nederlands
Nl flag.gif

Ελληνικά
Gr flag.gif

Hrvatska
Hr flag.gif

Česká republika
Cz flag.gif

Danmark
Dk flag.gif

Polska
Pl flag.png

România
Ro flag.png

Sverige
Se flag.gif