Two-Dimensional Brownian Motion

Timeline
Brownian motion
Distribution graph
Distribution graph

Description

The experiment consists of running a two-dimensional Brownian motion process \( \{(X_s, Y_s): 0 \le s \le t\} \) on the interval \( [0, t] \). Thus, \( \{X_s: s \in [0, \infty)\} \) and \( \{Y_s: s \in [0, \infty)\} \) are independent standard Brownian motions. The narrow graph at the top is the clock, and shows the passage of time. The position of the Brownian motion is shown as a red dot in the square graph on the left.

The random variables of interest are the coordinates \( X_t \) and \( Y_t \) at time \( t \). On each run, the values of these variables are recorded in the first table. The probability density functions and moments, and the empirical density functions and moments, are shown in the two distribution graphs and given in the two distribution tables. The parameter \( t \) can be varied with the input control.