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Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2008)Problem Set 6Due: October 22, 2008Topics: Joint PDFs, Conditional PDFs, Multiple continuous random variables, ,Derived distributions. [Text sections: 3.4-3.6, 4.1]1. Continuous random variables X and Y each take on experimental values between zero and one,with the joint pdf indicated below (the cutoff between probability density 0.8and1.6 occurs atx =0.5andy =0.5):0.51.6110.50.8 y(x,y)1 f X,Y(x,y)=0.8fX,Y(x,y)=1.6xyX,Y12-D View 3-D Viewxf(a) Are X and Y independent? Present a convincing argument for your answer.(b) Prepare neat, fully labelled plots for fX(x)andfY |X(y | 0.75).(c) Let R = XY and let A be the event X<0.5. Evaluate E[R | A].(d) Let W =min{X, Y } and determine the cumulative distribution function (CDF) of W .Youshould be able to reason out this part without doing any formal integrals.2. Signal Classification: A wire connecting two locations serves as the transmission medium forternary-valued messages; in other words, any transmitted message between locations is known tobe one of three possible symbols, each occurring with equal probability. It is also known that anynumerical value sent over this wire is subject to distortion; namely, if the value X is transmitted,the value Y received at the other end is described by Y = X + N where the random variableN represents additive noise, assumed to be normally distributed with mean μ = 0 and varianceσ2= 4. Assume that X is independent of N.(a) Suppose the transmitter encodes the three types of messages with the values −1, 0 and 1.At the other end, the received message is decoded according to the following rules:• If Y>12, then conclude the value 1 was sent.• If Y<−12. then conclude the value −1wassent.• If −12≤ Y ≤12, then conclude the value 0 was sent.Determine the probability of error for this encoding/decoding scheme. Reduce your calcu-lations to a single numerical value.Compiled October 7, 2008 Page 1 of 3Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2008)(b) Suppose that the scheme above is used and a value of Y =0.75 is received. What is theprobability that a 0 was transmitted?(c) In an effort to reduce the probability of error, the following modifications are made. Thetransmitter encodes the three types of messages with the values −2, 0 and 2 while thereceiver’s decoding rules are:• If Y>1, then conclude the value 2 was sent.• If Y<−1. then conclude the value −2wassent.• If −1 ≤ Y ≤ 1, then conclude the value 0 was sent.Repeat parts (a) and (b) for this modified encoding/decoding scheme.3. Alice and Bob work independently on a problem set. The time for Alice to complete the set is ex-ponentially distributed with mean 4 hours. The time for Bob to complete the set is exponentiallydistributed with mean 6 hours.(a) What is the probability that Alice finishes the problem set before Bob?(b) Given that Alice requires more than 4 hours, what is the probability that she finishes theproblem set before Bob?(c) What is the probability that one of them finishes the problem set an hour or more beforethe other one?(d) Let T be the time until the first one of them finishes the problem set. Find the PDF of T .(e) Alice and Bob work independently on 12 problem sets throughout the semester. The timethey take to complete problem sets is independent. Find the probability that Alice beatsBob (i.e. finishes a problem set before him in a particular week) exactly 7 of those weeks.4. Consider n independent tosses of a k-sided fair die. Let Xibe the number of tosses that resultin i. Find the covariance of X1and X2. Is your result positive or negative? Why?5. Consider the following problem and a purported solution. Either declare the solution to becorrect or explain the flaw.Question: Let X and Y have the joint densityfX,Y(x, y)=1,x∈ [0, 1] and y ∈ [x, x +1];0, otherwise.Find fX(x), fY(y), and fY |X(y|x). Are X and Y independent?Solution:fX(x)= fX,Y(x, y) dy = x+1x1 · dy =1.fY(y)= fX,Y(x, y) dx = 101 · dx =1.fY |X(y|x)=fX,Y(x, y)fX(x)=11=1.Since fY |X(y|x) does not depend on x,wehavethatX and Y are independent. Alternatively,X and Y are independent because fX,Y(x, y)=fX(x)fY(y).Compiled October 7, 2008 Page 2 of 3Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2008)6. An ambulance travels back and forth, at a known constant speed v, along a road of length .In other words, at any moment in time, consider the location of the ambulance to be uniformlydistributed over the interval (0, ), and it is equally likely to be traveling in either direction.Also at some moment in time, an accident (not involving the ambulance itself) occurs at a pointuniformly distributed on the road; that is, the accident’s distance from the starting end of theroad is also uniformly distributed over the interval (0, ). Assume the location of the accidentand the location of the ambulance are independent.(a) Supposing the ambulance is capable of immediate U-turns, compute the CDF and PDF ofthe ambulance’s travel time T to the location of the accident.(b)†Repeat part (a) but now suppose U-turns are only possible at either fixed end of the road.7.†As part of an advertising campaign, a chocolate company includes with each candy bar that itproduces a golden ticket with probability p, a silver ticket with probability q, and neither withprobability 1 − (p + q). All ticket placements are independent. If you collect both a golden ticketand a silver ticket, you win a prize that includes a trip to the factory and candy for life. Whatis the expected number of candy bars you have to purchase to win a prize?†Required for 6.431; optional for 6.041 Page 3 of


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MIT 6 041 - Problem Set 6

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