Beta Type-I Distribution
A continuous random variable $X$ is said to have a beta type-I distribution with parameters $m$ and $n$ if its p.d.f. is given by
$$ \begin{align*} f(x)&= \begin{cases} \frac{1}{B(m,n)}x^{m-1}(1-x)^{n-1}, & 0\leq x\leq 1;m,n>0 \\ 0, & Otherwise. \end{cases} \end{align*} $$
where $B(m,n) =\frac{\Gamma m \Gamma n}{\Gamma (m+n)}$. In notation, it can be written as $X\sim \beta_1(m,n)$.
Mean and Variance of Beta Type-I distiribution
The mean and variance of Beta Type-I distribution are $\mu_1^\prime =\frac{m}{(m+n)}$ and $\mu_2 = \frac{mn}{(m+n)^2(m+n+1)}$ respectuvely.
Proof
The mean of Beta Type-I distribution is
$$ \begin{eqnarray*} \text{mean = }\mu_1^\prime &=& E(X) \\ &=& \int_{0}^1 x\frac{1}{B(m,n)} x^{m-1}(1-x)^{n-1}\; dx\\ &=& \frac{1}{B(m,n)} \int_{0}^1 x^{m+1-1}(1-x)^{n-1}\; dx\\ &=& \frac{1}{B(m,n)} B(m+1, n)\\ &=& \frac{B(m+1, n)}{B(m,n)} \\ &=&\frac{\Gamma (m+1) \Gamma n}{\Gamma (m+n+1)}\cdot \frac{\Gamma(m+n)}{\Gamma m \Gamma n}\\ &=&\frac{m\Gamma m \Gamma n}{(m+n)\Gamma (m+n)}\cdot \frac{\Gamma(m+n)}{\Gamma m \Gamma n}\\ &=&\frac{m}{(m+n)} \end{eqnarray*} $$
To find variance of $X$, we need to find the $E(X^2)$.
$$ \begin{eqnarray*} \mu_2^\prime&= &E(X^2)\\ &=& \int_{0}^1 x^2\frac{1}{B(m,n)} x^{m-1}(1-x)^{n-1}\; dx\\ &=& \frac{1}{B(m,n)} \int_{0}^1 x^{m+2-1}(1-x)^{n-1}\; dx\\ &=& \frac{1}{B(m,n)} B(m+2, n)\\ &=& \frac{B(m+2, n)}{B(m,n)} \end{eqnarray*} $$
$$ \begin{eqnarray*} \mu_2^\prime &=&\frac{\Gamma (m+2) \Gamma n}{\Gamma (m+n+2)}\cdot \frac{\Gamma(m+n)}{\Gamma m \Gamma n}\\ &=&\frac{(m+1)m\Gamma m \Gamma n}{(m+n+1)(m+n)\Gamma (m+n)}\cdot \frac{\Gamma(m+n)}{\Gamma m \Gamma n}\\ &=&\frac{m(m+1)}{(m+n)(m+n+1)}. \end{eqnarray*} $$
Hence, the variance of Beta Type-I distribution is
$$ \begin{eqnarray*} \text{Variance = } \mu_2&=&\mu_2^\prime-(\mu_1^\prime)^2\\ &=&\frac{m(m+1)}{(m+n)(m+n+1)}-\bigg(\frac{m}{(m+n)}\bigg)^2\\ &=&\frac{m}{(m+n)}\bigg[\frac{m+1}{m+n+1}-\frac{m}{m+n}\bigg]\\ &=&\frac{m}{(m+n)}\bigg[\frac{m^2+mn+m+n -m^2-mn-m}{(m+n+1)(m+n)}\bigg]\\ &=&\frac{mn}{(m+n)^2(m+n+1)}. \end{eqnarray*} $$
Hence the mean and variance of Beta Type-I distribution are $\mu_1^\prime =\frac{m}{(m+n)}$ and $\mu_2 = \frac{mn}{(m+n)^2(m+n+1)}$ respectively.
Harmonic Mean
Let $H$ be the harmonic mean of Beta Type I distribution. Then the harmonic mean of Beta Type-I distribution is $H = \frac{m-1}{m+n-1}$.
Proof
$$ \begin{eqnarray*} \frac{1}{H} &=& E(1/X) \\ &=& \int_{0}^1 \frac{1}{x}\frac{1}{B(m,n)} x^{m-1}(1-x)^{n-1}\; dx\\ &=& \frac{1}{B(m,n)} \int_{0}^1 x^{m-1-1}(1-x)^{n-1}\; dx\\ &=& \frac{1}{B(m,n)} B(m-1, n)\\ &=& \frac{B(m-1, n)}{B(m,n)} \\ &=&\frac{\Gamma (m-1) \Gamma n}{\Gamma (m+n-1)}\cdot \frac{\Gamma(m+n)}{\Gamma m \Gamma n}\\ &=&\frac{\Gamma (m-1) \Gamma n}{\Gamma (m+n-1)}\cdot \frac{(m+n-1)\Gamma(m+n-1)}{(m-1)\Gamma (m-1) \Gamma n}\\ &=&\frac{m+n-1}{m-1}\\ \therefore H &=& \frac{m-1}{m+n-1}. \end{eqnarray*} $$
Hence the mean of Beta Type-I distribution is $H =\frac{m-1}{m+n-1}$.
Mode of Beta Type-I distribution
The mode of Beta Type-I distribution is $\frac{m-1}{m+n-2}$
Proof
The p.d.f. of Beta Type-I distribution is
$$ \begin{align*} f(x)&= \begin{cases} \frac{1}{B(m,n)}x^{m-1}(1-x)^{n-1}, & 0\leq x\leq 1;m,n>0 \\ 0, & Otherwise. \end{cases} \end{align*} $$
Taking $\log_e$, we get
$$ \begin{equation*} \log_e f(x)=-\log B(m,n) +(m-1)\log x+ (n-1)\log(1-x). \end{equation*} $$
Differentiating $\log f(x)$ w.r.t. $x$, we get
$$ \begin{eqnarray}\label{b11} \frac{d \log_e f(x)}{dx}& = & 0 +\frac{m-1}{x}+ \frac{n-1}{(1-x)}(-1)\\ & = & \frac{m-1}{x}- \frac{n-1}{(1-x)}. \end{eqnarray} $$
By the principal of maxima and minima,
$$ \begin{eqnarray*} & &\frac{d \log_e f(x)}{dx}=0 \\ \Rightarrow & & \frac{m-1}{x}- \frac{n-1}{(1-x)}=0\\ \Rightarrow & & (m-1)-x(m-1)= x(n-1)\\ \Rightarrow & & x(m+n-2) = (m-1)\\ \Rightarrow & & x_0=\frac{m-1}{m+n-2}. \end{eqnarray*} $$
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 $x= x_0 = \frac{m-1}{m+n-2}$.
Therefore, mode of Beta Type-I distribution is $\frac{m-1}{m+n-2}$.
Conclusion
In this tutorial, you learned about theory of Beta Type I distribution like the probability density function, mean, variance, harmonic mean and mode of Beta Type I distribution.
To read more about the step by step examples and calculator for Beta Type I distribution refer the link Beta Type I Distribution Calculator with Examples. This tutorial will help you to understand how to calculate mean, variance of Beta Type I distribution and you will learn how to calculate probabilities and cumulative probabilities for Beta Type I distribution with the help of step by step examples.
To learn more about other probability distributions, please refer to the following tutorial:
Let me know in the comments if you have any questions on Beta Type I Distribution and your thought on this article.
