Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2010)Problem Set 5Due October 18, 20101. Random variables X and Y are distributed according to the joint PDFfX,Y(x, y) =ax, if 1 ≤ x ≤ y ≤ 2,0, otherwise.(a) Evaluate the constant a.(b) Determine the marginal PDF fY(y).(c) Determine the expected value of1X, given that Y =32.2. Paul is vacationing in Monte Carlo. The amount X (in dollars) he takes to the casino eachevening is a r an dom variable with the PDF shown in the figure. At the end of each night, theamount Y that he h as on leaving the casino is uniformly distributed between zero and twicetheamount he took in.40fX(x )x (dollars)(a) Determine the joint PDF fX,Y(x, y). Be sure to indicate what the sample space is.(b) What is the probability that on any given night Paul makes a positive profit at the casino?Justify your reasoning.(c) Find and sketch the probab ility density function of Paul’s profit on any particular night,Z = Y − X. What is E [Z]? Please label all axes on your sketch.Page 1 of 3Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2010)3. X and Y are continuous random variables. X takes on values between 0 and 2 w hile Y takes onvalues between 0 and 1. Their joint pdf is indicated below.0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 200.20.40.60.81xyfX,Y(x, y) =32fX,Y(x, y) =12(a) Are X and Y independent? P resent a convincing argument for your answer.(b) Prepare neat, fully labelled plots for fX(x), fY |X(y | 0.5), and fX|Y(x | 0.5).(c) Let R = XY and let A be the event X < 0.5. Evaluate E[R | A].(d) Let W = Y − X and determine the cumulative distribution function (CDF) of W .4. Signal Classification: Consider the communication of binary-valued messages over sometransmission medium. Specifically, any message transmitted between locations is one of twopossible symbols, 0 or 1. Each symbol occurs 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 that is independent of X. The noise N is normally distributed w ithmean µ = 0 and variance σ2= 4.(a) Suppose the transmitter encodes the symbol 0 with the value X = −2 and the symbol 1with the value X = 2. At the other end, the received message is decoded according to thefollowing rules:• If Y ≥ 0, then conclude the symbol 1 was sent.• If Y < 0. then conclude the symbol 0 was sent.Determine the probability of err or for this encoding/decoding scheme. Reduce your calcu-lations to a single numerical value.(b) In an effort to reduce the probability of error, the following modifications are made. Thetransmitter encodes the symbols with a repeated scheme. The symbol 0 is encoded withthe vectorX = [−2, −2, −2]⊺and th e symbol 1 is encoded with the vector X = [2, 2, 2]⊺.The vectorY = [Y1, Y2, Y3]⊺received at the other end is described by Y = X + N. ThevectorN = [N1, N2, N3]⊺represents the noise vector where each Niis a random variableassumed to be normally distributed with mean µ = 0 and variance σ2= 4. Assume eachNiis independent of each other and independent of the Xi’s. Each component value of Yis decoded w ith the same rule as in part (a). The receiver then uses a majority rule todetermine which symbol was sent. The receiver’s d ecoding rules are:• If 2 or more components ofY are greater than 0, then conclude the symbol 1 was sent.• If 2 or more comp on ents ofY are less than 0, then conclude the symbol 0 was sent.Determine the probability of error for this modified encoding/decoding scheme. Reduceyour calculations to a single numerical value.Page 2 of 3Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2010)5. The random variables X and Y are described by a joint PDF which is constant within the unitarea quadrilateral with vertices (0, 0), (0, 1), (1, 2), and (1, 1).xy121 2(a) Are X and Y independent?(b) Find the marginal PDFs of X and Y .(c) Find the expected value of X + Y .(d) Find the variance of X + Y .6. A defective coin minting machin e produces coins whose probability of h eads is a random variableP with PDFfP(p) =1 + sin(2πp), if p ∈ [0, 1] ,0, otherwise.In essence, a specific coin produced by this machine will have a fixed probability P = p of givingheads, but you do not know initially what that probability is. A coin produced by this machineis selected and tossed repeatedly, with successive tosses assumed independent.(a) Find the probability that the first coin toss results in heads.(b) Given that the fir s t coin toss resulted in heads, find the conditional PDF of P .(c) Given that the first coin toss resulted in heads, find the conditional probability of heads onthe second toss.G1†. Let C be the circle {(x, y) | x2+y2≤ 1}. A point a is chosen randomly on the boundary of C andanother point b is chosen randomly from the interior of C (these points are chosen indepen dentlyand uniformly over their domains). Let R be the rectangle with sides parallel to the x- andy-axes with diagonal ab. What is the probability that no point of R lies outside of C?†Required for 6.431; optional for 6.041 Page 3 of
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