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ECE302 Spring 2006 Exam 1 February 27, 2006 Name: Solution Score: 1 /100 You must show ALL of your work for full credit. This exam is closed-book. One 8.5” by 11” page of notes is permitted. It should be turned in with the exam. Calculators may be used. 1. The random variable N has PMF PN (n) = ( c 2n 0 n ∈ {0, 1, 2, 3} otherwise (1) (a) What is the value of the constant c? The sum of the probabilities over all possible values of the random variable N must be one. Thus we calculate as follows. 1 1 1 1 c 0 + 1 + 2 + 3 =1 2 2 2 2 (2) So c = 8/15. (b) What is the probability that N 6= 0? P [N 6= 0] = 1 − P [N = 0] = 1 − 8/15 = 7/15 (3) (c) What is the probability that N is less than 2? P [N < 2] = P [N = 0] + P [N = 1] = 8 1 1+ 15 2 = 4 12 = 15 5 (4) ECE302 Spring 2006 Exam 1 2 February 27, 2006 2. Let the random variable C represent the amount the telephone company charges for calls to directory assistance. C is a function of the number R of phone numbers requested. Given that $0.50 R ∈ {1, 2} C= $0.75 R = 3 (5) $1.00 R = 4 And PR (r) = 0.6 0.2 R=1 R=2 R=3 R=4 0.1 0.1 (6) 0otherwise answer the following questions: (a) What is the expected number of requests E[R]? E[R] = X rPR (r) = 0.6(1) + 0.2(2) + 0.1(3) + 0.1(4) = 1.7 (7) r∈SR (b) What is the variance Var[R] of the number of requests? Var[R] = E[R2 ] − (E[R])2 = 2 (8) 2 2 2 0.6(1) + 0.2(2) + 0.1(3) + 0.1(4) = 3.9 − 2.89 = 1.01 − 1.7 2 (9) (10) (c) For calls that request more than one number, what is the conditional PMF PR|R>1 of the number of numbers requested? PR|R>1 = 1/2 r = 2 1/4 r ∈ {3, 4} 0 otherwise (11) (d) Given that a call requests more than one number, what is the expected number of requests E[R|R > 1]? 1 1 1 E[R|R > 1] = (2) + (3) + (4) = 2.75 2 4 4 (12) ECE302 Spring 2006 Exam 1 February 27, 2006 3 (e) What is the expected cost of a call to directory assistace? E[C] = 0.6(0.5) + 0.2(0.5) + 0.1(0.75) + 0.1(1) = $0.575 = 57.5 cents (13) (f) What is the standard deviation of a call to directory assistace? σC = q E[C 2 ] − (E[C])2 (14) so we need to find E[C 2 ], which is E[C 2 ] = 0.6(0.5)2 + 0.2(0.5)2 + 0.1(0.75)2 + 0.1(1)2 = 0.3563 (15) σC = 0.3563 − 0.3306 = 0.0257 (16) so ECE302 Spring 2006 Exam 1 February 27, 2006 4 3. You and a friend find a dime and a nickel. To decide who gets to keep them you flip the coins. If a coin comes up heads you win it; tails your friend wins it. (a) What is the sample space of the experiment? {Hh, Ht, T h, T t} (17) where H represents the dime landing heads up, h represents the nickle landing heads up, T represents the dime landing tails up, and t represents the nickle landing tails up. (b) Let S be the amount of money you win. What is the PMF of S? PS (s) = ( 1/4 s ∈ {0, 0.05, 0.10, 0.15} 0 otherwise (18) (c) What is the expected value of S? 1 1 1 1 E[S] = (0) + (0.05) + (0.10) (0.15) = $0.075 = 7.5 cents 4 4 4 4 (19) ECE302 Spring 2006 Exam 1 February 27, 2006 5 4. The CDF of random variable R is 0 0.1 (a) What is the PMF of R? r<2 2≤r<6 FR (r) = 0.7 6 ≤ r < 10 1 r ≥ 10 1/10 6/10 PR (r) = 3/10 0 r=2 r=6 r = 10 otherwise (20) (21) (b) What is the probability that −2.5 ≤ R < 8? −2.5 < 2 < 6 < 8 (22) P [−2.5 ≤ R ≤ 8] = 1/10 + 6/10 = 7/10 (23) so (c) What is the expected value of R? E[R] = 1 6 3 (2) + (6) + (10) = 6.8 10 10 10 (24) (d) What is the PMF of S = R2 ? PS (s) = 1/10 6/10 3/10 0 s=4 s = 36 s = 100 otherwise (25) (e) What is the expected value of S? E[S] = 6 3 1 (4) + (36) + (100) = 52 10 10 10 (26) ECE302 Spring 2006 Exam 1 6 February 27, 2006 5. A cellular phone call may be placed by a pedestrian or a driver. As the phone changes location along with its owner, a call may requires no handoffs (H0), one handoff (H1) or more than one handoff (H2) from base station to base station. We are given that the probability that a call comes from someone in a vehicle is 0.8. The probability that a call from a vehicle will not require a handoff is 0.5; the probability that it will require one handoff is 0.2. The probability that a call from a pedestrian will not require a handoff is 0.4; the probability that it will require one handoff is 0.1. (Note that this is not a very realistic model.) (a) What are the probabilities that calls from pedestrians, respectively drivers, will require two handoffs. (b) What is the probability that a call (of unknown origin) will not require a handoff? (c) When there is no handoff, what is the probability that the caller is in a vehicle? We are given that P [H0 |D] = 1/2 P [H0 |P ] = 4/10 P [H1 |D] = 2/10 P [H1 |P ] = 1/10 P [D] = 0.8 P [P ] = 0.2 (27) (a) P [H2 |P ] = 1 − P [H1 |P ] − P [H0 |P ] = 1 − 0.4 − 0.1 = 0.5 P [H2|D] = 1 − P [H1 |D] − P [H0 |D] = 1 − 0.5 − 0.2 = 0.3 (28) (29) (b) P [H0 ] = P [H0 |P ]P [P ] + P [H0 |D]P [D] = 0.4(0.2) + 0.5(0.8) = 0.48 (c) P [D|H0 ] = 0.5(0.8) 5 P [H0 |D]P [D] = = P [H0] 0.48 6 (30) (31) ECE302 Spring 2006 Exam 1 February 27, 2006 7 6. Suppose that a digit is selected at random from the set S = {−1, 0, 1, 2, 3, 4, 5} where the probability of selecting n is ( 0.2 n ∈ {−1, 0, 1} 0.1 n ∈ {2, 3, 4, 5} (32) A1 = {odd numbers in S} (33) A2 = {even numbers in S} (34) A3 = {n > 2} (35) A4 = {2, 3, 5} (36) A5 = {n < 4} (37) P [n] = Consider the following events (a) Find two independent events Ai and Aj or show that none exist. P [A1 ] P [A2 ] P [A3 ] P [A4 ] P [A5 ] = = = = = P [{−1, 1, 3, 5}] = 0.2 + 0.2 + 0.1 + 0.1 = 6/10 P [{0, 2, 4}] = 0.2 + 0.1 + 0.1 = 4/10 P [{3, 4, 5}] = 0.1 + 0.1 + 0.1 = 3/10 P [{2, 3, 5}] = 0.1 + 0.1 + 0.1 = 3/10 P [{−1, 0, 1, 2, 3}] = 0.2 + 0.2 + 0.2 + 0.1 + 0.1 = 8/10 P [A1 A2 ] = P [∅] = 0 P [A1 A3 ] = P [{3, 5}] = 0.1 + 0.1 = 0.2 P [A1 A4 ] = P [{3, 5}] = 0.1 + 0.1 = 0.2 P [A1 A5 ] = P [{−1, 1, 3}] = 0.2 + 0.2 + 0.1 = 0.5 P [A2 A3 ] = P [{4}] = 0.1 P [A2 A4 ] = P [{2}] = 0.1 P [A2 A5 ] = P [{0, 2}] = 0.2 + 0.1 = 0.3 P [A3 A4 ] = P [{3, 5}] = 0.1 + 0.1 = 0.2 P [A3 A5 ] = P [{3}] = 0.1 P [A4 A5 ] = P [{2, 3}] = 0.1 + 0.1 = 0.2 6 = 6= 6= 6 = 6= 6= 6= 6= 6= 6= (38) (39) (40) (41) (42) 0.24 = P [A1 ]P [A2 ](43) 0.18 = P [A1 ]P [A3 ](44) 0.18 = P [A1 ]P [A4 ](45) 0.48 = P [A1 ]P [A5 ](46) 0.12 = P [A2 ]P [A3 ](47) 0.12 = P [A2 ]P [A4 ](48) 0.32 = P [A2 ]P [A5 ](49) 0.09 = P [A3 ]P [A4 ](50) 0.24 = P [A3 ]P [A5 ](51) 0.24 = P [A4 ]P [A5 ](52) so no pair of events is independent. (b) For the events Ai and Aj you have chosen (or in general if none exist), what is the probability of {Ai ∩ Aj }? P [Ai ∩ Aj ] = P [Ai ]P [Aj ] i 6= j (53) (c) For the events Ai and Aj you have chosen (or in general if none exist), what is the probability of {Ai ∪ Aj }? P [Ai ∪ Aj ] = P [Ai ] + P [Aj ] − P [AiAj ] i 6= j (54) ECE302 Spring 2006 Exam 1 February 27, 2006 8 7. Short Questions (a) Suppose we are given that the PDF of the random variable X is fX (x) = e−bx + c for x ∈ [0, 1] and zero elsewhere. What two properties of PDF’s would you use in order to determine the acceptable values for the parameters b and c. (DON’T DO THE MATH. Just tell me the two facts.) Z ∞ −∞ fX (x) = 1 and fX (x) ≥ 0 ∀x (55) (b) A source wishes to transmit data packets to a receiver over a radio link. The receiver uses error detection to identify packets that have been corrupted by radio noise. When a packet is received error-free, the receiver sends an acknowledgment (ACK) back to the source. When the receiver gets a packet with errors, a negative acknowledgment (NAK) message is sent back to the source. Each time the source receives a NAK, the packet is retransmitted. We assume that each packet transmission is independently corrupted by errors with probability q. Would you model the distribution of the number of times the packet is transmitted by the source as Poisson (PX (x) below), Exponential (fX (x) below), or (c) Neither? (Circle the correct answer below and indicate the values of x for which the first option holds if you choose Poisson or Exponential. If you choose neither, suggest an alternative.) PX (x) = ( fX (x) = αx e−α x! 0 ( x∈ otherwise λe−λx x ∈ 0 otherwise Neither The point here was to realize that a random variable representing the number of times something happens must be a discrete random variable. I accepted either the Poisson or any other discrete probability mass distribution that made sense. The ranges of the values of x are x ∈ {0, 1, 2, . . .} for the Poisson distribution and x ∈ [0, ∞) for the Exponential. (c) What are the mean and the variance of a standard Gaussian (Normal) random variable? A standard normal random variable has mean 0 and variance 1. (d) How do you obtain a standard Gaussian (Normal) random variable from a nonstandard one? Let X be the nonstandard Gaussian random variable, µX be its mean, 2 and σX its variance. Then the random variable Z = (X − µX )/σX is a standard normal random variable. (e) The graph of the PDF of a Gaussian random variable X is symmetric about what value? the mean µX