2. Consider a random walk in the x-y plane. Suppose that the walker begins at the origin. In each time step, the

walker moves both up down and left/right. Let the random variables X be uniform over the interval -I f x $ 1. Let

the random variables Y, have the same distribution, except that it is along y direction. Assume that all X, and Y', are

independent. We wish to find the expectation value of the squared distance D that the random walker has travelled

after a steps.

(a) Find frix).

(b) Find fry).

(c) Find the joint PDF /(x, y).

(d) Find EX,I

(e) Find E[ Y].

(f) Find EX-1.

(g) Find ELY ].

(h) The squared distance that the walker has travelled after a steps is given by:

D = (Ex) + (Ex)

Using your results from (a) = (g), find E[D=].

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(i) Suppose instead that X, was a uniform random variable distributed over the interval 0 % x $ 1. How do your

above results change?