Mode of Beta Distribution of First kind and Beta Distribution of Second Kind

Mode of Beta Distribution of First Kind

The probability density function of beta distribution of first kind with parameter m and n is

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

Mode of this beta distribution of first kind is the solution of

f' (x) = 0 and f'' (x) < 0.

Differentiating f(x) with respect to x, we get

$$f '(x) \ = \ \frac{1}{\beta(m,n)} \ [{x^m}^{-1} \ (n-1) \ {(1-x)^n}^{-2} \ (-1) \ + \ {(1-x)^n}^{-1} \ (m-1) \ {x^m}^{-2} ]$$

$$= \ \frac{1}{\beta(m,n)} \ \left [{x^m}^{-2} \ {(1-x)^n}^{-1} \left ( (m-1) - \ \frac{(n-1)x}{1-x} \ \right ) \ \right ]$$

Making f ' (x) = 0, we get

$$(m-1) \ - \ \frac{(n-1)x}{1-x} \ = \ 0$$

$$\Rightarrow \ (m-1) \ (1-x) \ = \ (n-1) \ x$$

$$\Rightarrow \ x \ = \ \frac{m-1}{m+n-2} \ for \ m \ > 1 \ and \ n \ > \ 1.$$

It can show that at point

$$x \ = \ \frac{m-1}{m+n-2}$$

f''(x) < 0

Hence, the mode of the beta distribution of first kind with parameters m and n is

$$x = \ \frac{m-1}{m+n-2} \ \ for \ m \ > \ 1 \ and \ n \ > \ 1.$$

Also,

If m = 1, then x = 0 is the modal value.

If m > 1, n = 1 then x = 1 is the modal value.

If m < 1 then x = 0 is the mode.

If n <1 then x = 1 is the mode.

If m < 1, n < 1 then x = 0 and x = 1 are the modes of the beta distribution of first kind and hence in this situation, this distribution is bimodal.

If m = 1 and n = 1, then f(x) = 1 for 0 < x < 1. In this case, the mode of the beta distribution of first kind is each of x∈ (0,1). This is because, if m = 1 and n = 1 inβ1 (m,n), then the beta distribution of first kind reduces to rectangular distribution on (0,1)

Thus, the mode of the beta distribution of first kind depends on the values of the two parameters m and n.

Beta Distribution of Second kind

Definition:

A continuous random variable (rv) X is said to have a beta distribution of second kind with parameters m and n, if it has the probability density function (pdf)

$$f(x) = \ \frac{1}{\beta(m,n)} \ \frac{{x^m}^{-1}}{{(1+x)^m}^{+n}} \ \ : \ 0 \ < \ x \ < \ \infty$$

otherwise, 0.

Where m > 0, n > 0

If a random variable X follows a beta distribution of second kind with parameters m and n, then we write X ~β2 (m,n) or Beta2 (m,n).

It can be easily proved that the function f(x) is a density function, since,

$$\int_0^{\infty} \ f(x) \ dx \ = \ \frac{1}{\beta(m,n)} \\int_{0}^{\infty} \ \frac{{x^m}^{-1}}{{(1+x)^m}^{+n}} \ dx$$

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

$$= \ 1$$

Moments of Beta Distribution of Second Kind

Let X ~β2 (m,n). The rth moment about origin of X is given by

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

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

$$= \ \frac{1}{\beta(m,n)} \ \int_0^{\infty} \ \frac{{x^m}^{+r-1}}{{(1+x)^m}^{+n}} \ dx$$

$$= \ \frac{1}{\beta(m,n)} \ \int_0^{\infty} \ \frac{{x^(}^{m+r)-1}}{{(1+x)^(}^{m+r)+(n-r)}} \ dx$$

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

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

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

In particular,

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

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

$$= \ \frac{m}{n-1}$$

$$\therefore \ Mean = \ \frac{m}{n-1}$$

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

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

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

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

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

$$= \ \frac{m}{n-1} \ \left [ \ \frac{m+1}{n-2} \ - \ \frac{m}{n-1} \ \right ]$$

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

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

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

Mode of Beta Distribution of Second Kind

The beta distribution of second kind with parameters m and n has probability density function

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

The first derivative of f(x) with respect to x is

$$f ' (x) \ = \ \frac{1}{\beta(m,n)} \ \frac{{(1+x)^m}^{+n} \ (m-1) \ {x^m}^{-2} \ - \ {x^m}^{-1} \ (m+n) \ {(1+x)^m}^{+n-1}}{{(1+x)^2}^{(m+n)}}$$

$$= \ \frac{1}{\beta(m,n)} \ \frac{{x^m}^{-2} \ {(1+x)^m}^{+n-1} \ {(m-1) \ (1+x) \ - \ (m+n) x)}}{{(1+x)^2}^{(m+n)}}$$

$$= \ \frac{1}{\beta(m,n)} \ \frac{{x^m}^{-2} \ {m-1-(n+1)x}}{{(1+x)^m}^{+n+1}}$$

Making f'(x) = 0, we get

m - 1 - (n + 1) x = 0

$$\Rightarrow \ x \ = \ \frac{m-1}{n+1} \ \ \ for \ m \ > \ 1$$

It can also be shown that at

$$x \ = \ \frac{m-1}{n+1}$$

f''(x) < 0

When m = 1,

$$f(x) \ = \ \frac{1}{\beta (1,n) \ {(1+x)^1}^{+n}}$$

and f(x) has no maximum value on the range x≥ 0. However, at x = 0, the maximum value of f(x) on the range is

$$\frac{1}{\beta(1,n)}. \ \ when \ 0 \ < \ m \ < 1,$$

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

And f(x)→∞ as n→ 0 on the positive side. In this case, mode is given by x = 0.

Hence, the mode of the beta distribution of second kind is given by

$$x \ = \ \frac{m-1}{n+1} \ if \ m \ > \ 1.$$

and x = 0 if 0 < m < 1.

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.