Absolute Brownian Motion

Brownian motion
Distribution graph

Description

The experiment consists of running the absolute Brownian motion process \( \boldsymbol{U} = \{\left|X_s\right|: s \in [0, \infty) \} \) on the interval \( [0, t] \), where \( \boldsymbol{X} = \{X_s: s \in [0, \infty)\} \) is standard Brownian motion. This process is also referred to as Brownian motion reflected at 0. On each run, the path is shown in the graph on the left. The random variable of interest is the final position \( \left|X_t\right| \), which has the half-normal distribution with parameter \( \sqrt{t} \).

On each run, the value of the variable is recorded in the first table, and the point \( (t, \left|X_t\right|) \) is shown as a red dot in the path graph on the left. 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.