� Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) Problem Set 5 Due October 18, 2010 1. Random variables X and Y are distributed according to the joint PDF ax, if 1 ≤ x ≤ y ≤ 2,fX,Y (x, y) = 0, otherwise. (a) Evaluate the constant a. (b) Determine the marginal PDF fY (y). (c) Determine the expected value of1 X , given that Y = 32. 2. Paul is vacationing in Monte Carlo. The amount X (in dollars) he takes to the casino each evening is a random variable with the PDF shown in the figure. At the end of each night, the amount Y that he has on leaving th e casino is uniformly distributed between zero and twice the amount he took in. fX(x ) (a) Determine the joint PDF fX,Y (x, y). Be sure to indicate what the sample space is. (b) What is th e probability that on any given night Paul makes a positive profit at the casino? Justify your reasoning. (c) Find and sketch the probability density function of Paul’s profit on any particular night, Z = Y − X. What is E[Z]? Please label all axes on your sketch. 40 x (dollars) Page 1 of 3Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) 3. X and Y are continuous random variables. X takes on values between 0 and 2 while Y takes on values 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 2 0 0.2 0.4 0.6 0.8 1 y fX,Y (x, y) =3 2fX,Y (x, y) =1 2 x (a) Are X and Y independent? Present 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 fun ction (CDF) of W . 4. Signal Classification: Consider the communication of binary-valued messages over some transmission medium. Specifically, any message transmitted between locations is one of two possible symbols, 0 or 1. Each symb ol occurs with equal probability. It is also known that any numerical value sent over this wire is subject to distortion; namely, if th e value X is transmitted, the value Y received at the other end is described by Y = X + N where the random variable N represents additive noise that is independent of X. The noise N is normally distributed with mean µ = 0 and variance σ2 = 4. (a) Suppose the transmitter encodes the symbol 0 with the value X = −2 and the symbol 1 with the value X = 2. At the other en d, the received message is decoded according to the following 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 error f or this encoding/deco ding scheme. Reduce your calcu-lations to a s ingle numerical value. (b) In an effort to reduce the probability of error, the following modifications are made. The transmitter encodes the symb ols with a repeated scheme. The symbol 0 is en coded with the vector X = [−2, −2, −2]⊺ and the symb ol 1 is encoded w ith the vector X = [2, 2, 2]⊺ . The vector Y = [Y1, Y2, Y3]⊺ received at the other end is described by Y = X + N. The vector N = [N1, N2, N3]⊺ represents the noise vector where each Ni is a random variable assumed to be normally distributed w ith mean µ = 0 and variance σ2 = 4. Assume each Ni is independent of each other and independent of the Xi’s. Each component value of Y is decoded with the same rule as in part (a). The receiver then uses a majority rule to determine which symbol was s ent. The receiver’s decoding rules are: • If 2 or more components of Y are greater than 0, then conclude the symbol 1 was sent. • If 2 or more components of Y are less than 0, then conclude the symbol 0 was sent. Determine the probability of error for this modified encoding/decoding scheme. Red uce your calculations to a single numerical value. Page 2 of 3� Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) 5. The random variables X and Y are described by a joint PDF which is constant within the unit area quadrilateral with vertices (0, 0), (0, 1), (1, 2), and (1, 1). y x 1 2 1 2 (a) Are X and Y independent? (b) Find th e marginal PDFs of X and Y . (c) Find the expected value of X + Y . (d) Find th e variance of X + Y . 6. A defective coin minting machine produces coins whose probability of heads is a random variable P with PDF 1 + sin(2πp), if p ∈ [0, 1],fP (p) = 0, otherwise. In essence, a specific coin produ ced by this machine w ill have a fixed probability P = p of giving heads, but you do not know initially what that probability is. A coin produced by this machine is 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 first coin toss resulted in heads, find the conditional PDF of P . (c) Given that the first coin toss resulted in heads, find the conditional pr obability of heads on the second toss. G1† . Let C be the circle {(x, y) | x2 +y2 ≤ 1}. A point a is chosen randomly on the boundary of C and another point b is chosen randomly from the interior of C (these points are chosen independently and uniformly over their domains). Let R be the rectangle with sides parallel to the x- and y-axes with diagonal ab. What is the p robability that no point of R lies outside of C? †Required for 6.431; optional for 6.041 Page 3 of 3MIT OpenCourseWare http://ocw.mit.edu 6.041 / 6.431 Probabilistic Systems Analysis and Applied
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