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1. Extra Problems For MTH 511 Students2. Extra Challenge ProblemsMTH 411/511 PROBLEM SETThis problem was posed to me by Cameron Watt. As luck would have it, it matches this week’smaterial quite well. We consider a “random walk” in the plane where each step is one unit ata random angle. The walk begins at the origin. A random angle Θ1is chosen (let’s say from(π, π] but if you want to change it to [0, 2π), etc. that doesn’t change the problem), and thefirst “step” of the walk takes us from (0, 0) to (cos Θ1, sin Θ1). Then a second random angle Θ2is chosen, independent of the first, and the second step of the walk takes us from (cos Θ1, sin Θ1)to (cos Θ1+ cos Θ2, sin Θ1+ sin Θ2). Repeating this process we obtain four sequences of randomvariables:● (Θn)∞n=1which are independent uniform random variables on the interval [−π, π)● (Xn)∞n=1, where Xn=∑ni=1cos Θi● (Yn)∞n=1, where Yn=∑ni=1sin Θi● (Rn)∞n=1, where Rn=√X2n+Y2n.(1) The random variable X1is absolutely continuous. Find its density function.(2) Are X1and X2independent? Why or why not?(3) Are X1and Y1independent? Why or why not?(4) Do X1and Y1have an absolutely continuous joint distribution? Why or why not?(5) The random variables X1and X2have an absolutely continuous joint distribution. Findthe conditional density fX2∣X1(x2∣x1).(6) Find the joint density f (x1, x2).(7) Give an integral formula for the density fX2of X2. Can you do the integral? (Seriousquestion. So far, I can’t.)(8) Find P (R2≤ 1)(9) Find the cumulative distribution function FR2(t) of R2.(10) Do X2and Y2have an absolutely continuous joint distribution? If so, find the joint density.If not, explain why not.1. Extra Problems For MTH 511 StudentsNone this week.2. Extra Challenge ProblemsJust for fun. I have not thought enough about these problems to say exactly how hard they are–only enough to think that they are interesting.(1) Write the marginal density function fX2(x2) in closed form, or prove that it is impossibleto do so.(2) Find the cumulative distribution function FR3(t)(3) Find the marginal density fR3(r)(4) Find FRnand or fRnfor additional values of n.(5) Find a general formula for FRn(t) as a function of n and t and for fRn(r) as a function ofn and r.Date: November 14, 2020.1(6) Does the random walker tend to meander around while staying close to the origin, or doesit tend to wander away from the origin? Concretely, given a constant c, is limn→∞P (Rn<c) = 0? Does it depend on c?(7) What is E(Rn)?(8) What is limn→∞E(Rn)?(9) If limn→∞E(Rn) = ∞ (which is what I would guess .... but I’d be guessing) can you find afunction g(n) – the simpler the better – such that limn→∞E(Rn)/g(n) is a finite, nonzeroconstant?(10) If so, can you compute the

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