Problem $3.$ Consider a robot $($with negligible mass and volume$)$ initially located at the point $(0,0)$ in the

plane, and suppose that it undergoes a sequence of steps of fixed length, but in a completely random direction.

Specifically, suppose that the new position after each step is one unit of distance from the previous position

and at an angle of orientation from the previous position that is uniformly distributed over $(0,2).$

$($a$)$ Compute the expected square of the distance from the origin after $n$ steps.

$($b$)$ Simulate and plot $100$ trajectories of the robot for n$=20.$ You can use your favorite simulation tool $($e$.$g$.$

Matlab, Python, Julia, Java, C$+,$ etc$).$

$($c$)$ For each trajectory iin$\{1,2,$dots,$100\}$ generated in item $($b$),$ let $D_i^2$ be the square of the final distance

from the origin after the $n$ steps. Compute the sample mean of the variables $D_i^2,i=1,2,$dots,$100.$

The behavior described in this problem is called a random walk. For an application of random walks in

biological and navigation systems see this reference ${?}^1.$