Indefinite Integration of the Gamma Integral
and Related Statistical Applications

4.5 Beta Function

The general form of the beta function is defined by the following definite integral

$$\begin{equation*} {{B}_{x}}\left( \alpha ,\beta \right)=\int_{0}^{x}{{{t}^{\alpha -1}}{{\left( 1-t \right)}^{\beta -1}}dt}, \end{equation*}$$
Where $\alpha >0$ and $\beta >0$. When the upper limit of the integral $x$ is $1$, it is the complete beta function. Otherwise, it is the incomplete beta function. The ratio of the incomplete to complete beta function is the regularized beta function, which serves as the cumulative distribution function of many distributions, such as binomial distribution, beta distribution, and $F$ distribution.

The complete beta is closely associated with the gamma function. With some manipulations, we can transform the complete, incomplete, and regularized beta functions into a form of the "$h$" function

$$\begin{equation*} B\left(\alpha,\beta\right)=\frac{{\exp(-C)}h_{\alpha-1}^{-C}h_{\beta -1}^{-C}}{h_{\alpha +\beta -1}^{-C}},\tag{4.9} \end{equation*}$$
[Proof for (4.9)]

$$\begin{align*} \begin{split} {{B}_{x}}\left(\alpha,\beta\right)={{x}^{\alpha}}{\exp(-x)}h_{\alpha -1}^{-x}+\sum\limits_{i=1}^{\infty }&\Bigg\{ \left[ \prod\limits_{j=1,j\ne i}^{i+1}{\left( \beta -j \right)} \right] \\& \left[ {{x}^{\alpha }}{\exp(-x)}h_{\alpha -1}^{-x}-\sum\limits_{k=0}^{i-1}{{{\left( -1 \right)}^{k}}\frac{{{x}^{\alpha +k}}}{k!\left( \alpha +k \right)}} \right] \Bigg\}, \end{split} \tag{4.10} \end{align*}$$
[Proof for (4.10)]

$$\begin{equation*} {{I}_{x}}\left( \alpha ,\beta \right)=\frac{{{B}_{x}}\left( \alpha ,\beta \right)}{B\left( \alpha ,\beta \right)}. \end{equation*}$$

With the above results, we present the distribution functions of eight related distributions as an "$h$" function, such as the beta, binomial, $F$, beta-prime, negative binomial, Yule-Simon, noncentral $F$, and noncentral $t$ distributions.

C19 Beta distribution

$$\begin{equation*} F\left( x;\alpha ,\beta \right)={{I}_{x}}\left( \alpha ,\beta \right), \end{equation*}$$
where $\alpha >0$ (shape parameter), $\beta >0$ (shape parameter), and $x\in \left( 0,1 \right)$.

C20 Beta prime distribution

$$\begin{equation*} F\left( x;\alpha ,\beta \right)={{I}_{\tfrac{x}{1+x}}}\left( \alpha ,\beta \right), \end{equation*}$$
where $\alpha >0$ (shape parameter), $\beta >0$ (shape parameter), and $x\in \left( 0,1 \right)$.

C21 Binomial distribution

$$\begin{equation*} F\left( k;n,p \right)={{I}_{1-p}}\left( n-k,1+k \right), \end{equation*}$$
where $n\in \mathbb{N}$ (number of trials), $p\in \left[ 0,1 \right]$ (success probability in each trial), and $k\in \left\{ 0,1,\cdots ,n \right\}$.

C22 Negative binomial distribution

$$\begin{equation*} F\left( k;r,p \right)=1-{{I}_{p}}\left( k+1,r \right), \end{equation*}$$
where $r\in \mathbb{N}$ (number of failures until the experiment is stopped), $p\in \left( 0,1 \right)$ (success probability in each trial), and $k\in \left\{ 0,1,2,3,\cdots \right\}$.

C23 Yule-Simon distribution

$$\begin{equation*} F\left(k;\rho\right)=1-\frac{k{\exp\left(-C\right)}h_{k-1}^{-C}h_{\rho }^{-C}}{h_{k+\rho }^{-C}}, \end{equation*}$$
where $\rho >0$ (shape parameter) and $k\in \mathbb{N}$.

C24 F-distribution

$$\begin{equation*} F\left(x;{{d}_{1}},{{d}_{2}} \right)={{I}_{\tfrac{{{d}_{1}}x}{{{d}_{1}}x+{{d}_{2}}}}}\left( \frac{{{d}_{1}}}{2},\frac{{{d}_{2}}}{2} \right), \end{equation*}$$
where ${{d}_{1}}>0$ (degree of freedom), ${{d}_{2}}>0$ (degree of freedom), and $x\in \left[ 0,\infty \right)$.

C25 Noncentral F-distribution

$$\begin{equation*} F\left( x;{{d}_{1}},{{d}_{2}},\lambda \right)=\sum\limits_{i=0}^{\infty }{\left[ \frac{{{\left( \frac{1}{2}\lambda \right)}^{i}}{\exp\left(\tfrac{-\lambda }{2}\right)}}{i!} \right]}{{I}_{\tfrac{{{d}_{1}}x}{{{d}_{1}}x+{{d}_{2}}}}}\left( \frac{{{d}_{1}}}{2}+i,\frac{{{d}_{2}}}{2} \right), \end{equation*}$$
where ${{d}_{1}}>0$ (degree of freedom), ${{d}_{2}}>0$ (degree of freedom), $\lambda \ge 0$ (noncentrailty parameter), and $x\in \left[ 0,\infty \right)$.

C26 Noncentral t-distribution

$$\begin{equation*} F\left( x;\nu ,\delta \right)= \begin{cases} \Phi \left( -\delta \right)+\frac{{\exp\left(-\tfrac{1}{2}{{\delta }^{2}}\right)}}{2}\sum\limits_{j=0}^{\infty }{\frac{{{\left( \tfrac{{{\delta }^{2}}}{2} \right)}^{\tfrac{1}{2}j}}}{\Gamma \left( \tfrac{j}{2}+1 \right)}}{{I}_{\tfrac{{{x}^{2}}}{v+{{x}^{2}}}}}\left( \frac{j+1}{2},\frac{\nu }{2} \right) \quad \thinspace ,\text{if $x\ge 0$;}\\ \begin{split} \Phi \left( -\delta \right)&+\frac{{\exp\left(-\tfrac{1}{2}{{\delta }^{2}}\right)}}{2}\sum\limits_{j=0}^{\infty }{\frac{{{\left( \tfrac{{{\delta }^{2}}}{2} \right)}^{\tfrac{1}{2}j}}}{\Gamma \left( \tfrac{j}{2}+1 \right)}}{{I}_{\tfrac{{{x}^{2}}}{v+{{x}^{2}}}}}\left( \frac{j+1}{2},\frac{\nu }{2} \right)\\&-{\exp\left(-\tfrac{1}{2}{{\delta }^{2}}\right)}\sum\limits_{j=0}^{\infty }{\frac{{{\left( \tfrac{{{\delta }^{2}}}{2} \right)}^{j}}}{j!}}{{I}_{\tfrac{{{x}^{2}}}{v+{{x}^{2}}}}}\left( j+\frac{1}{2},\frac{\nu }{2} \right), \text{if $x<0$,} \end{split} \end{cases} \end{equation*}$$
where $v>0$ (degree of freedom), $\delta \in \mathbb{R}$ (noncentrailty parameter), and $x\in \left( -\infty ,\infty \right)$ (Johnson and Kotz 1970, p.205).

Welcome all math lovers!