Beta Distribution

The Beta Distribution is a continuous probability distribution defined on the interval [0, 1]. It has two positive shape parameters, \alpha and \beta. It has Probability Density Function given by

    \begin{equation*} f(x) = \frac{1}{B(\alpha, \beta)} x^{\alpha-1} (1 - x)^{\beta-1} \end{equation*}

where B(\alpha, \beta)=\frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\alpha + \beta)} is a normalizing constant insuring that f is in fact a probability density function, i.e. integrates to 1 and \Gamma is the Gamma function.

The beta distribution has been applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines. For a more thorough introduction to the beta distribution, consult the Beta Distribution page at Wikipedia.

To the right, I have plotted the Probabiltiy Density Function PDF and the corresponding Cumulative Density Function CDF for three different Beta Distributions. You can download the code creating these plots here.

Beta Cumulative Density Function CDF for different alpha and beta
Beta Probability Density Function PDF for different alpha and beta
Beta SAS Code Example

Below, I have written a small program that lets you set \alpha and \beta and draw the corresponding Probability Density Function for the Beta Distribution. I encourage you to play around with the shape and rate parameter and see how it affects the shape of the distribution. What happens when one is much larger than the other? And what happens when they are both large? Or small?

%let alpha=2;
%let beta=5;
 
data Beta_PDF;
      do x=0 to 1-0.01 by 0.01;
         pdf1=pdf('Beta', x, &alpha, &beta);
         output;
      end;
run;
 
title "Beta Densities";
title2 "For (*ESC*){unicode alpha}=&alpha and (*ESC*){unicode beta}=&beta";
proc sgplot data = Beta_PDF noautolegend;
   series x=x y=pdf1 / lineattrs=(thickness=3);
   yaxis label="PDF" labelattrs=(size=12 weight=Bold);
   xaxis label='x' labelattrs=(size=12 weight=Bold);
run;
title;