Beta Probabilities and Quantiles
The applet below displays the beta density function (alpha = 2, beta = 2) from Figure 4.17. Use the boxes to change alpha and beta to produce the other beta density functions illustrated in that figure.
Example 4.11
The applet below illustrates the solution to Example 4.11. Change x to 0.75 to determine the probability that at least 75% of her stock will be sold in a given week.
Exercise 4.131
This applet illustrates the solution to Exercise 4.131(a). The problem asks for the probability of less than .5, so the answer is 1 minus the probability illustrated in the applet.
Mathematical Details
The probability density function for the beta distribution is defined by
where
where
The probability corresponding to the tail is one minus the cumulative distribution function for the beta
distribution. Explicitly, the area or probability of the tail is given by
where the last term is known as the Incomplete Beta Function. This applet computes these values using
numerical methods.