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EE 416 Homework Assignment 1 Revised 1. (20 points) (a) Let A, B be events. Draw a Venn diagram that illustrates P (A ∪ B) = P (A) + P (B) − P (A ∩ B)? How does this equation simplify when (i) A and B are disjoint or (ii) when A and B are independent. For arbitrary A, B, C, extend this result to P (A ∪ B ∪ C) (b) If P (A) = a, P (B) = b > 0, then P (A | B) ≥ (a + b − 1)/b. Prove. (c) If P (A) = 1/3, P (B) = 1/4, can it be true that A ⊂ B? Prove. (d) For any N = 1, 2, 3, . . . , justify the “Union Bound,” P (A1 ∪ A2 ∪ . . . AN ) ≤ N X P (Ak ). k=1 First, work this out for N = 2 and N = 3. Then use induction on N . When will this bound be a good approximation? When will it be a useless approximation? 2. (20 points) (a) This is a problem in geometric probability, and you need the Table.A dime, penny, etc have the given diameters. Play a game, where we toss each type of coin, each in turn, at random onto a huge infinitely large square grid. This grid has lines of width W = 5mm and each are separated by a distance D, as measured from inside-to-inside of the lines. Coin Roosevelt Dime Lincoln Cent Jefferson Nickel Washington Quarter Kennedy Half Dollar Eisenhower Dollar Diameter 17.9 mm 19.0 mm 21.2 mm 24.3 mm 30.6 mm 38.1 mm Sketch a picture of the grid and label. You toss the coin and win if you do not touch a grid line.If we want the odds to be 50; 50, what should the Distance D be, given the diameters and widths are specified as above. Get a general formula, and evaluate for each coin in the list. 3. (10 points) (b)Geometric Probability. It is equally likely that two signals arrive at a receiver at any time 0 < t < T , where T is known. Assume that the receiver is jammed if both signals arrive within time difference τ . Sketch the sample space and by computing areas on a T ×T square, (i) find a general formula for the probability of jamming, PJ , and (ii) the probability that both signals are received clearly, PC . Show (iii) that for very small γ := τ /T, PJ ≈ 2γ. As T → ∞, PJ → 0. Explain (iv) why this is true. 1 4. (10 points) A communications link is given, connecting the source, S, to the receiver, R. It consist of N identical series elements a, followed by M parallel elements b, as depicted. Each element a, b fails independently of the others. Let Ak be the event that ak is working, with probability P (Ak ) = pk , and similarly for Bk , with P (Bk ) = qk . Using unions and intersections of appropriate events, (i) find a general expression for the event O, that the link is out (outage). Let A denote the event that the series link is working and B that the parallel link is working. Then (ii) set pk = 0.99, qk = 0.50, and find and set M so that P (B) = 0.99. and find PO , the corresponding probability of outage, for 1 ≤ N ≤ 10, (iii) how large must N be to assure that PO ≤ 0.1 ? (iv) What happens if N > 10 ? Solve with MATLAB. 5. (40 points) We discussed the Birthday Problem in class. Birthdays are randomly distributed over the year, D = 365 days. How many people P must you query so that the probability of two or more birthdays matching exceeds 0.5. Write a Monte Carlo Simulation of this problem, by looping over T = 1000 independent trials. Show that your result agrees with the theory. Extend your simulation to estimate the probability that two or more birthdays either match or are adjacent. This is a common problem in uncoordinated access to a shared medium. The people represent nodes trying to access slots (the days). If two or more nodes try and access the same slot, a collision occurs and they must retry. The question is the maximum density P/D that allows uncoordinated access. To draw P people’s birthdays randomly on N days use b = 1 +floor(rand(P,1)*N); The flow is knt = 0; for t = 1:T Generate the Brithdays of p people in [1:N] See if any birthdays satisfy the conditions, if so set knt = knt+1; end pest = knt/T; 2