Beta distribution

Definition of the Beta distribution

A first example : order statistics of the uniform  distribution

Using only very simple arguments, we showed that the distribution of the k^th order statistic of the uniform distribution in [0, 1] is :

where n is the sample size.

It is often convenient to express factorials as Gamma functions, and owing to the property of the Gamma function :

Γ(m) = (m - 1)!

when m is an integer, the above expression becomes

 with :

    * α = k

    * β = n - k + 1

Here, α and β are integers, but the above expression still makes mathematical sense if this condition is relaxed and if α and β are only constrained to be positive real numbers, but it is then not clear that f_α,β (x) represents a probability density function anymore. This would be the case only if :

that is if

for all pairs of positive real numbers (α, β).

This is indeed the case as will be shown in the Tutorial below.

Relationship between the Beta function and the Gamma function

The integral

defines a function of α and β known as the Beta function. So what we will show is that the Beta and the Gamma functions entertain the following relationship

In what follows, we'll dispense with the Beta function and use this relationship between the Beta and the Gamma functions to express all results in terms of the Gamma function only.

Gamma(1/2)

We'll show that an immediate consequence of this important equation is :

The Beta distribution

So we have identified a family of probability distributions that are continuous in [0, 1] and indexed by two real positive parameters α and β. These distributions are collectively called the Beta distribution, that will be denoted Beta(α, β).

The probability density function of the Beta distribution is :

Animation

This animation illustrates the Beta distribution.

Slide the cursors and explore the various shapes of the Beta distribution.

    1) When both α and β are larger than 1, the Beta distribution is unimodal, and its density is 0 at both ends of the range.

    2) When α is larger than 1 but β is smaller than 1, the density fuction is monotonically increasing from 0 with a vertical asymptote at x = 1 (with the opposite result when α is smaller than 1 and β larger than 1.

    3) When α = 1 and β is larger than 1, the density is a monotonically decreasing power function of x that intersects the x = 0 axis (with the opposite result when β = 1 and α is larger than 1).

Note that the density function is linear when α = 1 and β = 2.

    4) When α = 1 and β is smaller than 1, the density is monotonically increasing with the x = 1 axis being an asymptote (with the opposite result when β = 1 and α is smaller than 1).

    5) The density is U-shaped when both α and β are smaller than 1. Both vertical axes are asymptotes.

    6) The density is uniform in [0, 1] when both α and β are equal to 1.

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When α = β, the Beta distribution is symmetrical about x = .5 (click on "Lock").

Properties of the Beta distribution

Switching parameters

It is clear that

Beta_{β, α}(x) = Beta_{α, β} (1 - x) 

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We'll establish the following properties of the Beta distribution.

Moments of all orders

The Beta distribution has moments of all orders given by:

Mean

The mean µ of the Beta distribution is :

Variance

The variance σ² of the Beta distribution is :

Mode

We leave it to the reader to show that when both α and β are larger than 1, the mode of the Beta distribution is :

Sufficient statistics

We'll identify sufficient statistics for α, for β, as well as for the pair (α, β) when the values of both these parameters are unknown.

We'll then show that these statistics are in fact minimal sufficient.  

Transforming to a Beta distribution

The Beta distribution is related :

    * To the Gamma distribution,

    * And to Fisher's F distribution.

Relationship between the Gamma distribution and the Beta distribution

Let :

    * X ~ Gamma(α, θ)

    * Y ~ Gamma(β, θ)

be two independent Gamma distributed r.v. with the same value θ for the second parameter.

We'll show that

Relationship between Fisher's F distribution and the Beta distribution

Let F be a r.v. distributed as Fisher's F_{n ,m} .

We'll show that

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In this Tutorial, we first establish the relationship between the Gamma and the Beta functions. This will justify the claim that the expression we proposed for the Beta distribution is indeed a probability density function. All further results will be expressed in terms of the Gamma function.

We'll show that as an immediate consequence :

a result frequently used in many areas of statistics. 

We then establish the basic properties of the Beta distribution. This will be remarkably easy owing to the particular structure of the pdf of the Beta distribution.

We finally establish the relationship between :

    * The Gamma and the Beta distributions as stated above. The demonstration will also show that (X + Y ) and X / (X + Y ) are independent, an unexpected result. As a bonus, we'll discover again the addivity property of the Gamma distribution.

    * Fisher's F distribution and the Beta distribution, as stated above.

THE BETA DISTRIBUTION

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