### Normal Estimation Experiment

#### Description

The experiment is to generate a random sample \(\boldsymbol{X} = (X_1, X_2, \ldots, X_n)\) from the normal distribution with mean \(\mu\) and standard deviation \(\sigma\). 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}
M &= \frac{1}{n} \sum_{i=1}^n X_i \\
S^2 &= \frac{1}{n-1} \sum_{i=1}^n (X_i - M)^2 \\
T^2 &= \frac{1}{n} \sum_{i=1}^n (X_i - M)^2 \\
W^2 &= \frac{1}{n} \sum_{i=1}^n (X_i - \mu)^2
\end{align}
The statistic \(M\) is the sample mean and is a point estimator of the distribution mean \(\mu\). The statistics \(S^2\), \(T^2\), and \(W^2\) are sample variances and are point estimators of the distribution variance \(\sigma^2\). (For \(W^2\), we must assume that \(\mu\) is known.) The true bias and mean square error for each estimator is shown in the table for that estimator. On each update, the empirical bias and mean square error of each estimator are also recorded in these tables. The parameters \(\mu\), \(\sigma\), and \(n\) can be varied with the input controls.