Beta Type-II Distribution

Beta Type-II distribution

A continuous random variable $X$ is said to have a beta type-II distribution with parameters $m$ and $n$ if its p.d.f. is given by

$$ \begin{align*} f(x)&= \begin{cases} \frac{1}{B(m,n)}\cdot\frac{x^{m-1}}{(1+x)^{m+n}}, & 0\leq x\leq\infty;m,n>0 \\ 0, & Otherwise. \end{cases} \end{align*} $$

where $B(m,n) =\frac{\Gamma m \Gamma n}{\Gamma (m+n)}$. In notation, it can be written as $X\sim \beta_2(m,n)$.

Mean and Variance of Beta Type-II Distribution

The mean and variance of Beta Type-II distribution are $\mu_1^\prime=\frac{m}{n-1}$ and $\mu_2 =\frac{m(m+n-1)}{(n-1)^2(n-2)}$ respectively.

Proof

The mean of Beta Type-II distribution is

$$ \begin{eqnarray*} \text{mean = }\mu_1^\prime &=& E(X) \\ &=& \int_{0}^\infty x\frac{1}{B(m,n)}\cdot\frac{x^{m-1}}{(1+x)^{m+n}} \; dx\\ &=& \frac{1}{B(m,n)} \int_{0}^\infty \cdot\frac{x^{m+1-1}}{(1+x)^{(m+1)+(n-1)}}\; dx\\ &=& \frac{1}{B(m,n)} B(m+1, n-1)\\ &=& \frac{B(m+1, n-1)}{B(m,n)} \\ &=&\frac{\Gamma (m+1) \Gamma (n-1)}{\Gamma (m+n)}\cdot \frac{\Gamma(m+n)}{\Gamma m \Gamma n}\\ &=&\frac{m\Gamma m \Gamma (n-1)}{\Gamma (m+n)}\cdot \frac{\Gamma(m+n)}{\Gamma m (n-1)\Gamma (n-1)}\\ &=&\frac{m}{n-1}. \end{eqnarray*} $$

To find variance, we need to find $E(X^2)$.

$$ \begin{eqnarray*} \mu_2^\prime&= &E(X^2)\\ &=& \int_{0}^\infty x\frac{1}{B(m,n)}\cdot\frac{x^{m-1}}{(1+x)^{m+n}} \; dx\\ &=& \frac{1}{B(m,n)} \int_{0}^\infty \cdot\frac{x^{m+2-1}}{(1+x)^{(m+2)+(n-2)}}\; dx\\ &=& \frac{1}{B(m,n)} B(m+2, n-2)\\ &=& \frac{B(m+2, n-2)}{B(m,n)} \\ \end{eqnarray*} $$

$$ \begin{eqnarray*} \text{i.e., } \mu_2^\prime &=&\frac{\Gamma (m+2) \Gamma (n-2)}{\Gamma (m+n)}\cdot \frac{\Gamma(m+n)}{\Gamma m \Gamma n}\\ &=&\frac{(m+1)m\Gamma m \Gamma (n-2)}{\Gamma (m+n)}\cdot \frac{\Gamma(m+n)}{\Gamma m (n-1)(n-2)\Gamma (n-2)}\\ &=&\frac{m(m+1)}{(n-1)(n-2)}. \end{eqnarray*} $$

Hence,the variance of Beta Type-II distribution is

$$ \begin{eqnarray*} \text{Variance = } \mu_2&=&\mu_2^\prime-(\mu_1^\prime)^2\\ &=&\frac{m(m+1)}{(n-1)(n-2)}-\bigg(\frac{m}{n-1}\bigg)^2\\ &=&\frac{m}{(n-1)}\bigg[\frac{m+1}{n-2}-\frac{m}{n-1}\bigg]\\ &=&\frac{m}{(n-1)}\bigg[\frac{mn-m+n -1-mn+2m}{(n-1)(n-2)}\bigg]\\ &=&\frac{m(m+n-1)}{(n-1)^2(n-2)}. \end{eqnarray*} $$

Thus the mean and variance of Beta Type-II distribution are $\mu_1^\prime=\frac{m}{n-1}$ and $\mu_2 =\frac{m(m+n-1)}{(n-1)^2(n-2)}$ respectively.

Harmonic Mean of Beta Type-II Distribution

Let $H$ be the harmonic mean of Beta Type II distribution. Then the harmonic mean of Beta Type-II distribution is $H =\frac{m-1}{n}$.

Proof

$$ \begin{eqnarray*} \frac{1}{H} &=& E(1/X) \\ &=& \int_{0}^\infty \frac{1}{x}\frac{1}{B(m,n)}\cdot\frac{x^{m-1}}{(1+x)^{m+n}} \; dx\\ &=& \frac{1}{B(m,n)} \int_{0}^\infty \frac{x^{m-1-1}}{(1+x)^{m-1+n+1}} \; dx\\ &=& \frac{1}{B(m,n)} B(m-1, n+1)\\ &=& \frac{B(m-1, n+1)}{B(m,n)}\\ &=&\frac{\Gamma (m-1) \Gamma (n+1)}{\Gamma (m+n)}\cdot \frac{\Gamma(m+n)}{\Gamma m \Gamma n}\\ &=&\frac{\Gamma (m-1) n\Gamma n}{\Gamma (m+n)}\cdot \frac{\Gamma(m+n)}{(m-1)\Gamma (m-1) \Gamma n}\\ &=&\frac{n}{m-1}\\ \end{eqnarray*} $$

Hence the harmonic mean of Beta Type-II distribution is

$$ \begin{equation*} \therefore H = \frac{m-1}{n}. \end{equation*} $$

Mode of Beta Type-II distribution

The mode of Beta Type-II distribution is $\frac{m-1}{n+1}$.

Proof

The p.d.f. of Beta Type-II distribution is $\frac{m-1}{n+1}$

$$ \begin{align*} f(x)&= \begin{cases} \frac{1}{B(m,n)}\cdot\frac{x^{m-1}}{(1+x)^{m+n}} , & 0\leq x\leq \infty; m,n>0 \\ 0, & Otherwise. \end{cases} \end{align*} $$

Taking $\log_e$, we get

$$ \begin{equation*} \log_e f(x)=-\log B(m,n) +(m-1)\log x- (m+n)\log(1+x). \end{equation*} $$

Differentiating $\log f(x)$ w.r.t. $x$, we get

$$ \begin{eqnarray*} \frac{d \log_e f(x)}{dx}& = & 0 +\frac{m-1}{x}- \frac{m+n}{(1+x)}\\ & = & \frac{m-1}{x}- \frac{m+n}{(1+x)}. \end{eqnarray*} $$

By the principal of maxima and minima,

$$ \begin{eqnarray*} & &\frac{d \log_e f(x)}{dx}=0 \\ \Rightarrow & & \frac{m-1}{x}- \frac{m+n}{(1+x)}=0\\ \Rightarrow & & (m-1)+x(m-1)= x(m+n)\\ \Rightarrow & & x(n+1) = (m-1)\\ \Rightarrow & & x_0=\frac{m-1}{n+1}. \end{eqnarray*} $$

Also,

$$ \begin{equation*} \frac{d^2 \log_e f(x)}{dx^2}\bigg|_{x=x_0}< 0. \end{equation*} $$
Hence, $f(x)$ is maximum at $x= x_0 = \frac{m-1}{n+1}$.

Therefore, mode of Beta Type-II distribution is $\frac{m-1}{n+1}$.

Reference

Refer Beta Type II Distribution Calculator is used to find the probability density and cumulative probabilities for Beta Type II distribution with parameter $\alpha$ and $\beta$.

Beta Type II Distribution Calculator

Let me know in the comments if you have any questions on Beta Type-II Distribution and what your thought on this article.

VRCBuzz co-founder and passionate about making every day the greatest day of life. Raju is nerd at heart with a background in Statistics. Raju looks after overseeing day to day operations as well as focusing on strategic planning and growth of VRCBuzz products and services. Raju has more than 25 years of experience in Teaching fields. He gain energy by helping people to reach their goal and motivate to align to their passion. Raju holds a Ph.D. degree in Statistics. Raju loves to spend his leisure time on reading and implementing AI and machine learning concepts using statistical models.

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