The experiment consists of running the Brownian motion process \( \boldsymbol{X} = \{ X_s: s \in [0, \infty) \} \) with drift parameter \( \mu \) and scale parameter \(\sigma\) on the interval \( [0, t] \). The process \( \boldsymbol{X} \) can be constructed at \( X_s = \mu s + \sigma Z_s \) where \( \boldsymbol{Z} = \{Z_s: s \in [0, \infty)\} \) is standard Brownian motion. On each run, the path is shown in red in the graph on the top. The graph of the mean function \( m(t) = \mu t \) is shown in blue. The random variable of interest is the final position \( X_t \), which has the normal distribution with mean \( \mu t \) and standard deviation \( \sigma \sqrt{t} \).
On each run, the value of the variable is recorded in the data table, and the point \( (t, X_t) \) is shown as a red dot in the path graph. The probability density function and moments, and the empirical density function and moments, are shown in the distribution graph and in the distribution table. The parameters \( t \), \( \mu \), and \(\sigma\) can be varied with input controls.