### Standard Brownian Motion

Brownian motion
Distribution graph

#### Description

The experiment consists of running the standard Brownian motion process $$\boldsymbol{X} = \{X_s: s \in [0, \infty) \}$$ on the interval $$[0, t]$$. On each run, the path is shown in the graph on the left. There are three random variables of interest:

• The final position $$X_t$$, which has the normal distribution with mean 0 and standard deviation $$\sqrt{t}$$.
• The maximum position $$Y_t = \max\{X_s: s \in [0, t]\}$$ which has the half-normal distribution with parameter $$\sqrt{t}$$.
• The last zero $$Z_t = \max\{s \in [0, t]: X(s) = 0\}$$ which has the arcsine disstribution on the interval $$[0, t]$$.

On each run, the value of each variable is recorded in the first table, and the points $$(t, X_t)$$, $$(0, Y_t)$$, and $$(Z_t, 0)$$ are shown as red dots in the path graph on the left. Any of the three variables can be selected with the list box, and then the probability density function and moments, and the empirical density function and moments, are shown in the distribution graph on the right and given in the distribution table on the right. The parameter $$t$$ can be varied with the input control.