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
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