Beta Type-I Distribution

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:

Probability distributions

Let me know in the comments if you have any questions on Beta Type I Distribution and 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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