09 Beta distribution

09 Beta distribution - < it skewed to the right (see...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Beta distribution The beta density function is defined over the interval 0 x 1 and it can be used to model proportions (e.g. the proportion of time a machine is under repair, the proportion of a certain impurity in a chemical product, etc.). The probability density function of the beta distribution is given by: f ( x ) = x α - 1 (1 - x ) β - 1 B ( α,β ) , α > 0 , β > 0 , 0 x 1 . where, B ( α,β ) = Z 1 0 x α - 1 (1 - x ) β - 1 dx. The shape of the distribution depends on the values of the parameters α and β . When α = β the distribution is symmetric about 1 2 as shown in the figure below: 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 x f(x) Beta distribution densities with parameters α = β α = 10 α = 3 α = 1 α = 1 4 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
When α > β the distribution is skewed to the left and when
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: < it skewed to the right (see next gure). 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 x f(x) Beta distribution densities with parameters and < = 6 , = 4.5 = 1.5 , = 3 Even though x was dened in the interval 0 x 1, its use can be extended to random variables dened over some nite interval, c x d . In this case we can simply rescale the variable using y = x-c d-c , and y will be between 0 and 1. It can be shown that B ( , ) = ( )( ) ( + ) . Using this relation between the beta and gamma functions we can nd the mean and variance of the beta distribution: E ( X ) = + and var ( X ) = ( + ) 2 ( + + 1) . 2...
View Full Document

This note was uploaded on 01/14/2011 for the course STATS 100A taught by Professor Wu during the Fall '07 term at UCLA.

Page1 / 2

09 Beta distribution - < it skewed to the right (see...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online