The experiment consists of running the standard Brownian motion procees \( \boldsymbol{X} = \{X_s: s \in [0, \infty)\} \) and the integrated Brownian motion process \( \boldsymbol{Y} = \{Y_s: s \in [0, \infty) \} \) on the interval \( [0, t] \), so that \( Y_s = \int_0^s X_u \, du \). On each run, the paths of \( \boldsymbol{X}\) and \(\boldsymbol{Y}\) are shown in the first graph, in green and red respectively. The random variables of interest are the final positions \(X_t\) and \( Y_t \). The first has the normal distribution with mean 0 and standard deviation \(\sqrt{t}\), while the second has the normal distribution with mean 0 and standard deviation \( \sqrt{t^3 / 3} \).
On each run, the value of the variables are recorded in the data table, and the points \((t, X_t)\) and \( (t, Y_t) \) are shown as green and red dots on the appropriate paths. The probability density function and moments, and the empirical density function and moments, are shown in the distribution graphs and given in the distribution tables. The parameter \( t \) can be varied with the input control.