# AP Statistics Curriculum 2007 Beta

### From Socr

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'''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 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 |

- | :<math>\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> | + | :<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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:<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=== |

## Revision as of 20:54, 11 July 2011

### 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