# AP Statistics Curriculum 2007 Beta

### From Socr

(→Beta Distribution) |
(→Beta Distribution) |
||

Line 18: | Line 18: | ||

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 |

## Revision as of 20:53, 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