### The Beta-Binomial Experiment

Timeline
Beta Distribution graph
Beta-Binomial Distribution graph

In the beta-binomial experiment, a random probability $$P$$ has a beta distribution with left parameter $$a$$ and right parameter $$b$$. Given $$P = p$$, we conduct $$n$$ Bernoulli trials with success probability $$p$$. Random variable $$Y$$ is the number of successes, and has the beta-binomial distribution with parameters $$n$$, $$a$$, and $$b$$. The values of $$P$$ and $$Y$$ are recorded on each run, and the outcomes of the trials are shown in the timeline graph (red for success and green for failure). The probability density function of $$P$$ is shown in the first graph, and the probability density function of $$Y$$ is shown in the second graph and is given in the second table. On each run, the value of $$P$$ is shown in the first graph, and the empirical probability density function of $$Y$$ is shown in the second graph and recorded in the second table. Also recorded are $$M = Y / n$$, the proportion of successes and $$Z = (a + Y) / (a + b + n)$$. Random variable $$Z$$ is the Bayesian estimator of the success parameter $$p$$ when $$p$$ is modeled by $$P$$. As a function of $$n$$, it is a martingale. The parameters $$n$$, $$a$$, and $$b$$ can be varied with input controls.