### Beta Estimation Experiment

#### Description

The experiment is to generate a random sample \(\boldsymbol{X} = (X_1, X_2, \ldots, X_n)\) from the beta distribution with left parameter \(a\) and right parameter \(b\). The probability density function is shown in blue in the graph, and on each update, the sample density function is shown in red. On each update, the following statistics are recorded:
\begin{align}
U & = \frac{M(M - M_2)}{M_2 - M^2} \\
V & = \frac{(1 - M)(M - M_2)}{M^2 - M^2} \\
U_b & = b \frac{M}{1 - M} \\
V_a & = a \frac{1 - M}{M} \\
W & = - \frac{n}{\sum_{i-1}^n \ln(X_i)}
\end{align}
where \(M = \frac{1}{n} \sum_{i=1}^n X_i\) and \(M_2 = \frac{1}{n} \sum_{i=1}^n X_i^2\)

Statistics \(U\) and \(V\) are estimators of \(a\) and \(b\), respectively. Statistic \(U_b\) is an estimator of \(a\) assuming that \(b\) is known, while \(V_a\) is an estimator of \(b\) assuming that \(a\) is known. Statistics \(W\) is an estimator of \(a\), assumping that \(b = 1\). On each update, the empirical bias and mean square error of each estimator are recorded in the table for that estimator. The parameters \(a\), \(b\), and \(n\) can be varied with the input controls.