Introducing the Beta Prime Family

The Beta Prime distribution, also known as the inverted Beta, arises naturally when taking the ratio of two independent Gamma random variables. If $X$ and $Y$ follow Gamma distributions with shapes $\alpha$ and $\beta$ respectively, then the random variable $Z=\frac{X}{Y}$ has a Beta Prime distribution. This distribution is supported on the positive real line, exhibiting heavy tails when the parameter $\beta$ is small. Its probability density function is written

$f\left(z\right)=\frac{z^{\alpha -1}{1+z}^{-\alpha -\beta }}{B\left(\alpha ,,,\beta \right)}$ for $z>0$,

where the normalizing constant $B\left(\alpha ,,,\beta \right)$ is the Beta function. The Beta function can be expressed in terms of the Gamma function as $B\left(\alpha ,,,\beta \right)=\frac{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{\Gamma }$. That relationship provides the foundation for evaluating probabilities analytically.

The cumulative distribution function is the integral of the density from zero up to a specified $z$. While there are series expansions for this integral, a straightforward approach is to perform numerical integration. The calculator below employs Simpson’s rule over a fine partition to approximate the CDF, trading speed for accuracy. The heavy-tailed nature means more samples may be needed when $\beta$ is small, but for moderate values convergence is rapid.

Why Study Beta Prime?

Beyond its definition as a ratio of Gamma variables, the Beta Prime distribution surfaces in Bayesian statistics as the conjugate prior for a variance parameter. It also finds use in reliability engineering and finance when modeling the distribution of certain risk ratios. Compared with the simpler Beta distribution, the Beta Prime extends to infinite support, making it suitable when the variable of interest has no natural upper bound. By adjusting $\alpha$ and $\beta$, one can obtain shapes ranging from sharply peaked to gradually decaying.

For example, when $\alpha =2$ and $\beta =3$, the distribution has a mode near $\frac{\alpha -1}{\beta }$, providing an asymmetric bell shape. Decreasing $\beta$ stretches the right tail dramatically, demonstrating heavy-tailed behavior. Such flexibility explains why Beta Prime appears in actuarial science when modeling extreme claims or in biology when describing certain growth ratios.

Understanding the Beta Prime distribution builds intuition for other related families. When $\alpha$ and $\beta$ grow large, the distribution approaches a log-normal form under suitable scaling. Conversely, when $\beta$ approaches one while $\alpha$ remains moderate, the distribution approximates a Pareto law. These connections highlight the Beta Prime’s role as a bridge between common probability models.

Exploring the Math

At its core, the Beta Prime distribution emerges from a transformation of variables in multivariate calculus. If $X~\mathrm{Gamma}\left(\alpha ,,,1\right)$ and $Y~\mathrm{Gamma}\left(\beta ,,,1\right)$ are independent, the change of variables $Z=\frac{X}{Y}$ and $W=Y$ yields a joint density where integration over $W$ results in the Beta Prime density for $Z$. The heavy tail arises because $Z$ can become arbitrarily large when $Y$ takes small values, so the density decays polynomially rather than exponentially.

To compute probabilities, we often rely on the incomplete Beta function $B\left(z,;,\alpha ,,,\beta \right)$. Although not elementary, this function is implemented in many scientific libraries. When unavailable, numerical integration is a practical substitute. Simpson’s rule approximates the integral by dividing the interval into an even number of slices, fitting quadratic curves to pairs of slices, and summing the resulting areas. The error decreases rapidly with the number of subdivisions, making it suitable for interactive tools.

Keep in mind that the Beta Prime parameters must be positive. When $\alpha$ or $\beta$ is less than one, the density becomes steep near the origin, producing a strong mode at zero. Conversely, larger parameters flatten the curve. By moving sliders or entering values manually, you can see how the distribution evolves. This experimentation fosters understanding far beyond reading formulas alone.

Using the Calculator

Enter the shape parameters $\alpha$ and $\beta$ along with a point $x$. After pressing the Compute button, the script evaluates the PDF and numerically integrates to find the CDF. If you prefer, you can adjust the number of subintervals inside the code to trade accuracy for speed. The output displays both values with six decimal places, providing a quick reference for statistics or probability exercises. Try comparing the Beta Prime to the more familiar Beta distribution by considering the transformation $Y=\frac{X}{1+X}$; this mapping converts a Beta Prime variable $X$ into a Beta variable $Y$.

The Beta Prime distribution reveals how rich probability theory can be. Even a seemingly simple ratio of Gamma variables results in a flexible model that bridges heavy-tailed and light-tailed phenomena. By experimenting with the calculator and delving into the MathML formulas above, you’ll gain a deeper appreciation of special functions like the Beta and Gamma functions, as well as numerical techniques for evaluating them. Whether you work in Bayesian inference, reliability, or any field involving positive-valued random variables, this distribution offers a versatile tool for modeling and analysis.