# AP Statistics Curriculum 2007 Beta

### From Socr

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+ | ==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Beta Distribution== | ||

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===Beta Distribution=== | ===Beta Distribution=== | ||

'''Definition''': Beta distribution is a distribution that models events which are constrained to take place within an interval defined by a minimum and maximum value. | '''Definition''': Beta distribution is a distribution that models events which are constrained to take place within an interval defined by a minimum and maximum value. | ||

- | <br />'''Probability density function''': For | + | <br />'''Probability density function''': For <math>X\sim \operatorname{Beta}(\alpha,\beta)\!</math>, the Beta probability density function is given by |

:<math>\frac{x^{\alpha-1}(1-x)^{\beta-1}}{\Beta(\alpha,\beta)}</math> | :<math>\frac{x^{\alpha-1}(1-x)^{\beta-1}}{\Beta(\alpha,\beta)}</math> | ||

where | where | ||

- | *<math>\alpha</math> is a positive shape parameter | + | *<font size="3"><math>\alpha</math></font> is a positive shape parameter |

- | *<math>\beta</math> is a positive shape parameter | + | *<font size="3"><math>\beta</math></font> is a positive shape parameter |

- | *<math>\Beta(\alpha,\beta)=\int_0^1 t^{\alpha-1}(1-t)^{\beta-1}dt</math> or | + | *<math>\textstyle\Beta(\alpha,\beta)=\int_0^1 t^{\alpha-1}(1-t)^{\beta-1}dt</math> or <br /><math>\textstyle\Beta(\alpha,\beta)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}</math>, where <math>\Gamma(k)!=(k-1)!=1 \times 2 \times\ 3 \times\cdots \times (k-1)</math> |

- | + | ||

*x is a random variable | *x is a random variable | ||

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where | where | ||

- | *<math>\Beta_x(\alpha,\beta)=\int_0^x t^{\alpha-1}(1-t)^{\beta-1}dt</math> | + | *<math>\textstyle\Beta_x(\alpha,\beta)=\int_0^x t^{\alpha-1}(1-t)^{\beta-1}dt</math> |

- | *<math>\Beta(\alpha,\beta)=\int_0^1 t^{\alpha-1}(1-t)^{\beta-1}dt</math> | + | *<math>\textstyle\Beta(\alpha,\beta)=\int_0^1 t^{\alpha-1}(1-t)^{\beta-1}dt</math> |

<br />'''Moment generating function''': The Beta moment-generating function is | <br />'''Moment generating function''': The Beta moment-generating function is | ||

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:<math>Var(X)=\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}</math> | :<math>Var(X)=\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}</math> | ||

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===Applications=== | ===Applications=== | ||

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*The time it takes to complete a task | *The time it takes to complete a task | ||

*The proportion of defective items in a shipment | *The proportion of defective items in a shipment | ||

- | |||

===Example=== | ===Example=== | ||

- | Suppose that DVDs in a certain shipment are defective with a Beta distribution with | + | Suppose that DVDs in a certain shipment are defective with a Beta distribution with <font size="3"><math>\alpha=2</math></font> and <font size="3"><math>\beta=5</math></font>. Compute the probability that the shipment has 20% to 30% defective DVDs. |

We can compute this as follows: | We can compute this as follows: | ||

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The figure below shows this result using [http://socr.ucla.edu/htmls/dist/Beta_Distribution.html SOCR distributions] | The figure below shows this result using [http://socr.ucla.edu/htmls/dist/Beta_Distribution.html SOCR distributions] | ||

<center>[[Image:Beta.jpg|600px]]</center> | <center>[[Image:Beta.jpg|600px]]</center> | ||

+ | |||

+ | <hr> | ||

+ | * SOCR Home page: http://www.socr.ucla.edu | ||

+ | |||

+ | {{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Beta}} |

## Current revision as of 22:35, 18 July 2011

## Contents |

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

### Beta Distribution

**Definition**: Beta distribution is a distribution that models events which are constrained to take place within an interval defined by a minimum and maximum value.

**Probability density function**: For , the Beta probability density function is given by

where

- α is a positive shape parameter
- β is a positive shape parameter
- or

, where - x is a random variable

**Cumulative density function**: Beta cumulative distribution function is given by

where

**Moment generating function**: The Beta moment-generating function is

**Expectation**: The expected value of a Beta distributed random variable x is

**Variance**: The Beta variance is

### Applications

The Beta distribution is used in a range of disciplines including rule of succession, Bayesian statistics, and task duration modeling. Examples of events that may be modeled by Beta distribution include:

- The time it takes to complete a task
- The proportion of defective items in a shipment

### Example

Suppose that DVDs in a certain shipment are defective with a Beta distribution with α = 2 and β = 5. Compute the probability that the shipment has 20% to 30% defective DVDs.

We can compute this as follows:

The figure below shows this result using SOCR distributions

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

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