Beta Distribution

Beta distribution is one of the two parameters continuous probability distribution. A continuous distribution in which a beta function is involved is known as a beta distribution. A beta function, is denoted byβ, is defined as

$$\beta (m,n) \ = \ \int_{0}^{1} \ {x^m}^{-1} \ {(1-x)^n}^{-1} \ dx$$

where m > 0, n > 0.

The relation between beta and gamma function is

$$\beta(m,n) \ = \ \frac{\Gamma m \ \Gamma n}{\Gamma(m + n)} $$

There are two beta functions

1. beta function of first kind
2. beta function of second kind

Beta Distribution of First Kind

Definition

A random variable X is said to have a beta distribution of first kind with parameters m and n if its probability density function (pdf) is given by

$$f(x) \ = \ f(x; \ m, \ n) \ = \ \frac{{x^m}^{-1} \ {(1-x)^n}^{-1}}{\beta(m,n)} \ \ \ \ ; \ 0 \ < \ x \ < \ 1.$$

0; otherwise.

where, m > 0, n > 0.

If a random variable X follows a beta distribution of first kind with parameters m and n, then we write it as

$$X \ ∼ \beta_1 \ (m, \ n) \ or \ Beta_1 \ (m, \ n).$$

The above function f(x) is a probability density function (pdf) since,

$$\int_{0}^{1} \ f(x) \ dx \ = \ 1.$$

This can be easily proved as

$$\int_{0}^{1} \ f(x) \ dx \ = \ \int_{0}^{1} \ \frac{{x^m}^{-1} \ {(1-x)^n}^{-1}}{\beta(m,n)} \ dx$$

$$= \ \frac{1}{\beta(m, \ n)} \ \int_{0}^{1} \ {x^m}^{-1} \ {(1-x)^n}^{-1} \ dx$$

$$= \ \frac{1}{\beta(m, \ n)} \ \beta(m, \ n)$$

$$= \ 1$$

The shape of the beta curve depends on the values of m and n. When the parameters m and n are equal, the beta curve y = f(x) is symmetrical about 1/2. If m > n, the beta curve is skewed to the left and if m < n, it is skewed to the right. The graphical presentation of beta distribution for different values of m and n is given below:

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The distribution function of the beta distribution is given by

$$F(x) \ = \ 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ; \ x \ < \ 0.$$

$$= \ \int_{0}^{x} \ \frac{{u^m}^{-1} \ {(1-u)^n}^{-1}}{\beta(m, \ n)} \ du \ \ ; \ 0 \ < x \ < \ 1.$$

$$= \ 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ; \ x \ > 1.$$

This f(x) is also called incomplete beta function.

Remark:

If m = n = 1, then the beta distribution reduces to a uniform distribution over the interval [0, 1]. That is, if X ~ β₁ (m, n) then X ~ U (0, 1) when m = 1 and n = 1.

Moments of Beta Distribution of First Kind

Let ~β₁(m, n). The r^th moment about origin of beta distribution of first kind is given by

$$\mu_{r}^{'} \ = \ E(X^r)$$

$$= \ \int_{0}^{1} \ X^r \ f(x) \ dx$$

$$= \ \frac{1}{\beta(m, \ n)} \ \int_{0}^{1} \ {x^(}^{m+r)-1} \ {(1-x)^n}^{-1} \ dx$$

$$= \ \frac{1}{\beta(m, \ n)} \ \beta(m+r,n)$$

$$= \ \frac{\Gamma(m+n) }{\Gamma m \ \Gamma n} \ . \ \frac{\Gamma(m+r) \ \Gamma n}{\Gamma(m+r+n)}$$

$$= \ \frac{\Gamma(m+n) \ \Gamma(m+r)}{\Gamma m \ \Gamma(m+r+n)} \ \ \ \ \ ; \ r \ = \ 1, \ 2, \ 3, \ 4$$

In particular,

$$\mu_1^{'} \ =\ \frac{\Gamma(m+n) \ \Gamma(m+1)}{\Gamma m \ \Gamma(m+1+n)}$$

$$= \ \frac{\Gamma(m+n) \ m \ \Gamma(m)}{\Gamma m \ (m + n) \ \Gamma(m+n)}$$

$$= \ \frac{m}{m + n}$$

$$\therefore \ Mean \ = \ \frac{m}{m+n}$$

$$\mu_2^{'} \ = \\frac{\Gamma(m+n) \ \Gamma(m+2)}{\Gamma m \ \Gamma(m+2+n)}$$

$$= \ \frac{\Gamma(m+n) \ (m+1) \ m \ \Gamma m}{\Gamma m \ \ (m+n+1) \ \ (m+n) \ \Gamma(m+n)}$$

$$= \ \frac{m \ (m+n)}{(m+n+1) (m+n)}$$

$$\therefore \ Variance, \mu_2 \ = \ \mu_2^{'} \ - \ (\mu_1^{'})^2$$

$$= \ \frac{m (m+1)}{(m+n+1) \ (m+n)} \ - \ \left ( \frac{m}{m+n} \ \right )^2$$

$$= \ \frac{m}{m+n} \ \left [ \ \frac{m+1}{m+n+1} \ - \ \frac{m}{m+n} \ \right ]$$

$$= \ \frac{m}{m+n} \ \left [ \ \frac{(m+1) \ (m+n) \ - \ m(m+n+1)}{(m+n+1) \ (m+n)} \ \right ]$$

$$= \ \frac{m (m^2 + mn + m + n - m^2 - mn -m)}{(m+n)^2 \ (m + n +1)}$$

$$\Rightarrow \ Var \ (X) \ = \ \frac{mn}{(m+n)^2 \ (m+n+1)}$$

In this manner, we can obtain the other central momentsμ₃,μ₄ and then the measures of skewness and kurtosis of the beta distribution as follows:

The third cental moment:

$$\mu_3 \ = \ \frac{2mn \ (n-m)}{(m+n)^3 \ (m+n+1) \ (m+n+2)}$$

The fourth central moment:

$$\mu_4 \ = \ \frac{3mn \ {mn \ (m+n-6) \ + \ 2(m+n)^2}}{(m+n)^4 \ (m+n+1) \ (m+n+2) \ (m+n+3)}$$

The moment coefficient of skewness:

$$\beta_1 \ = \ \frac{4(m-n)^2 \ (m+n+1)}{mn (m+n+2)^2}$$
The moment coefficient of kurtosis:

$$\beta_2 \ = \ \frac{3(m+n+1) \ {mn (m+n-6) \ + \ 2(m+n)^2}}{mn(m+n+2) \ (m+n+3)}$$

Bibliography

Sukubhattu N.P. (2013). Probability & Inference - II. Asmita Books Publishers & Distributors (P) Ltd., Kathmandu.

Larson H.J. Introduction to Probability Theory and Statistical Inference. WileyInternational, New York.