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1 UNIVERSITY OF DUBLIN TRINITY COLLEGE FACULTY OF ENGINEERING, MATHEMATICS & SCIENCE SCHOOL OF ENGINEERING Electronic & Electrical Engineering Junior Sophister Engineering Trinity Term, 2015 Annual Examinations Probability Modelling (3E3) Date: 16th May 2015 Venue: EXAMINATION HALL Time: 14:00 16:00 Anthony Quinn ANSWER QUESTION 1, and any THREE of the remaining five questions. Question 1 is worth 30 marks in toto. All remaining questions are worth equal marks (i.e. 70/3 marks each). The percentage division of marks within each question is indicated on the paper. Permitted Materials: Calculator Graph paper Page 1 of 8

2 Q.1 [Compulsory] Answer ALL the following questions. (a) Consider three propositions, A, B and C, such that Pr[A] = Pr[B] = Pr[C] = 0.3. A and B are mutually exclusive. Also, C is independent of both A and B. Evaluate Pr[A B C]. (b) A digital source generates a sequence of independent bits, such that Pr[1] = p. The most probable number of 1 s in any block of 5 bits emanating from this source is 5. Deduce the possible range of values of p. (c) Each node in a particular network can be in one of four states, with probabilities 0.1, 0.6, 0.2 and 0.1 respectively, Assuming the nodes are independent, evaluate the probability that among any five nodes in the network three are in state 2 and one is in state 3. (d) Particles impinge uniformly on a circular detector of radius 5 cm, but none can do so within a radius of 2 cm of the centre of the detector. Deduce and sketch the cumulative distribution function (cdf) of R (cm), the distance of any such impinging particle from the centre of the detector. (e) A plant manufactures a particular medical device, and measures the time to failure of a sample of these. 50% of the sample are found to have failed within 2 years. Under the usual assumption, evaluate the probability that any one device manufactured by this plant will fail within one year. Page 2 of 8

3 Q.2 (a) Define conditional probability, justifying its defining equation, and use this to deduce Bayes rule. [40 %] (b) The transition probability diagram of a ternary (three-state) communication channel is illustrated in Figure Q ɛ 0 ɛ 0 1 ɛ 1 ɛ 2 ɛ 1 1 ɛ 2 Figure Q The probability of a 1 at the source is 0.5, with the other two source symbols being equiprobable. The conditional probabilities of error at the receiver as defined in the Figure are ɛ 1 = 0.1, ɛ 0 = ɛ 2 = 0.2. Adopt a repetition coding scheme at the transmitter, where each source symbol is repeated 3 times. Propose a suitable decoding scheme at the receiver, to combat the uncertainty (noise) introduced by this channel, and hence evaluate the total probability of an error at the output of your decoder. [60 %] Page 3 of 8

4 Q.3 (a) Define a homogeneous Markov chain (HMC), in terms of its transition probability matrix, T, and its initial probability vector, p 1. [25 %] (b) Prove that the k-step-ahead transition probability matrix of the HMC is T k, k = 1, 2, 3,... [25 %] (c) Each image in a sequence of electrocardiogram (ECG) images of the mitral valve of a patient is labelled by experts as O (open) or C (closed) or A (anomalous). The following sequence is recorded: O O C C C O A C C A A O O O C A C O O A C Expert 1 and 2 each adopt a HMC model for the sequence. Expert 2, alone, also assumes that the HMC is colour-blind. Estimate each of the following: (i) Expert 1 s probability that exactly one anomaly (A) will occur in the next two ECG images that follow the sequence above. Hint: assign each transition probability in proportion to the number of times the transition is observed in the sequence. [35 %] (ii) Expert 2 s probability that an anomaly (A) will occur in an ECG image in the longrun. [15 %] Page 4 of 8

5 Q.4 (a) The cumulative distribution function of any random variable is non-decreasing. Define the two phrases in italics, and provide a proof of this statement. [50 %] (b) The dynamic range of an electrical voltage signal is ( 12, +12] V. This signal forms the input to a quantizer which maps each sampled value of the signal to a B + 1-bit binary word, B 1. Carefully specifying all the assumptions that you make, design a minimal value for B, such that the signal-to-noise ratio (SNR) at the output of the quantizer is at least 40 db. [50 %] Page 5 of 8

6 Q.5 (a) Define the correlation coefficient, ρ XY, of any two random variables, X and Y, and state the range of possible values of ρ XY. [30 %] (b) The voltages, V 1 and V 2, measured at two points in an electrical system are jointly normal (Gaussian), with means 0 V and 2 V respecively, and variances 1 V 2 and 2 V 2 respectively. The correlation coefficient of these voltages is (i) Evaluate the probability that V 1 is positive, given that V 2 is measured as 3 V. (ii) Deduce the equation of the regression line of V 2 as a function of V 1 = v 1. Note on the bivariate Gaussian (normal) distribution: [40 %] [30 %] If where with then where m = [ mx [ X Y m Y ] ] N (m, Σ),, Σ = σ 2 X σ XY σ XY = ρ XY σ X σ Y, σ XY σ 2 Y f(x Y = y) = N (m X y, σ 2 X y),, m X y = m X + ρ XY σ X σ Y (y m Y ), σ 2 X y = σ 2 X ( 1 ρ 2 XY ). Page 6 of 8

7 Q.6 (a) The voltages transduced by a biomedical sensor are thought to be independently distributed, either uniformly or normally. Among a large number of such recorded voltages, the following two statistics are noted: (i) their average is 1.2 Volts, and (ii) 65% of them are positive. (i) Using these statistics, estimate the variance of the recorded voltages under each of the two assumed models. (ii) Propose an informal statistical test to choose between the two models. [30 %] [15 %] (b) The time intervals between cars passing a point on a rural road are measured in seconds (to one decimal place). The following are among eight such measurements: Adopting an appropriate parametric probability model, estimate the probabiity that among the next 6 such measurements two will be more than 10 seconds, and three will be less than 5 seconds. [55 %] Page 7 of 8

8 Page 8 of 8 c University of Dublin 2015

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