# AP Statistics Curriculum 2007 Exponential

(Difference between revisions)
 Revision as of 18:46, 11 July 2011 (view source)TracyTam (Talk | contribs) (→Exponential Distribution)← Older edit Current revision as of 22:33, 18 July 2011 (view source)JayZzz (Talk | contribs) (7 intermediate revisions not shown) Line 1: Line 1: + ==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Exponential Distribution== + ===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. '''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($\lambda$), the exponential probability density function is given by +
'''Probability density function''': For $X\sim \operatorname{Exponential}(\lambda)\!$, the exponential probability density function is given by :$\lambda e^{-\lambda x}\!$ :$\lambda e^{-\lambda x}\!$ Line 8: Line 10: where where *e is the natural number (e = 2.71828…) *e is the natural number (e = 2.71828…) - *$\lambda$ is the mean time between events + *$\lambda is the mean time between events *x is a random variable *x is a random variable Line 17: Line 19: where where *e is the natural number (e = 2.71828…) *e is the natural number (e = 2.71828…) - *[itex]\lambda$ is the mean time between events + *$\lambda is the mean time between events *x is a random variable *x is a random variable Line 31: Line 33: :[itex]Var(X)=\frac{1}{\lambda^2}$ :$Var(X)=\frac{1}{\lambda^2}$ - ===Applications=== ===Applications=== Line 43: Line 44: *The height of various molecules in a gas at a fixed temperature and pressure in a uniform gravitational field *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 *The monthly and annual maximum values of daily rainfall and river discharge volumes - ===Example=== ===Example=== Suppose you usually get 2 phone calls per hour. Compute the probability that a phone call will arrive within the next hour. 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: + 2 phone calls per hour means that we would expect one phone call every 1/2 hour so $\lambda=0.5. We can then compute this as follows: :[itex]P(0\le X\le 1)=\sum_{x=0}^1 0.5e^{-0.5x}=0.393469$ :$P(0\le X\le 1)=\sum_{x=0}^1 0.5e^{-0.5x}=0.393469$ Line 54: Line 54: The figure below shows this result using [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html SOCR distributions] The figure below shows this result using [http://socr.ucla.edu/htmls/dist/Exponential_Distribution.html SOCR distributions]
[[Image:Exponential.jpg|600px]]
[[Image:Exponential.jpg|600px]]
+ + +

## 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…)
• 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

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

### 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 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 λ = 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