ECE 302: Probabilistic Methods in Electrical Engineering

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1 ECE 302: Probabilistic Methods in Electrical Engineering Test I : Chapters 1 3 3/22/04, 7:30 PM Print Name: Read every question carefully and solve each problem in a legible and ordered manner. Make sure you write down all your answers without skipping details. I don t give credit for wrong answers and partial credit will be given only when sufficient details have been provided. 1. Experiments, Outcomes, and Sample Spaces: Several experiments are described below. Note that in specifying an experiment, we first specify an operation or procedure to be carried out and then specify the method of observation (i.e., outcomes or sample space such as S 1 = {x x R and x > 0} and S 2 = {a,e,i,o,u}). For the following experiments, identify the sample spaces (be very specific, use proper definitions of sets). (8 points): (a) E 1 : Toss a true coin and observe the up face. S 1 : (b) E 2 : Toss a true coin three times and observe the sequence of heads and tails. S 2 : (c) E 3 : Toss a coin three times and observe the total number of heads. S 3 : (d) E 4 : Toss a pair of dice and observe the sum of the up faces. S 4 :

2 i 2. Union and Intersection of Events, Probabilities, and Independence: An electronic system consists of four components: A, B, C, and D. The operation of the system can be represented by four switches, with A and B in series, and C and D in parallel with the series combination of A and B, as shown in Figure 1 below. Continuity between input and output means the system is in operation. Assume that C and D are 95% pairwise A B C D Figure 1: Electronic circuit for Problem 2. reliable, but A and B are 99% pairwise reliable, with all failures occurring independently. (12 points). (a) Represent the event O = {Entire system is operating}, in terms of the events A B C D = {A is operating} = {B is operating} = {C is operating} = {D is operating}. (b) Find the overall reliability of the system (i.e., the probability that the system is operating).

3 ii 3. Discrete Random Variables, Independence, and Conditional Probabilities: Consider a 4-bit register that stores a Binary Coded Decimal (BCD) number, with all numbers from 0 to 9 equally likely. The BCD code is shown in Table 1 below. The power supply current is i = k µa, where k is the number of 1 s in the register. (15 points). Table 1. BCD Code for Problem 3. No. B 4 B 3 B 2 B (a) Which pair of bits, if any, are independent? (b) Find P[B 4 = 1 B 1 = 1]. (c) Find the probability that the power supply is in the range 0.20 µa i 0.30 µa.

4 iii 4. Total Probability and Bayes Theorem: Three printers do work for the publications office of IUPUI. The publications office does not negotiate a contract penalty for late work, and the data below reflect a large amount of experience with these printers. Printer, i Fraction of Contracts Fraction of Time Delivery Held by Printer i More than One Month Late A department observes that its recruiting booklet is more than a month late, what is the probability that the contract is held by printer 3? (15 points).

5 iv 5. Bernoulli Trials: A certain class of highway bridge has 12 supporting columns, each of which is inspected annually for weakness. Experience shows that, with annual inspection, the probability of a column weakening seriously during the year is 10 4, independent of the condition of the other columns. What is the probability that two bad columns would occur (10 points).

6 v 6. Discrete Random Variables: An integer is chosen at random between 0 and 4 inclusive (i.e., i is selected from the set i = {0,1,2,3,4} at random). A random variable is assigned to each outcome according to the following rule X = sin ( ) iπ, 4 where i is the outcome. Find and sketch the probability mass function (PMF) for X, P X (x). (15 points).

7 vi 7. Continuous Random Variables: Suppose the probability density function (PDF)of X, f X (x), is f X (x) = ( 0, 0 x < 2 a ) 2 (x 2), 2 x < 4 3a ( a 2) x, 4 x 5 0, x > 5 Determine the following: (15 points). (a). The value of a that makes f X (x) a valid PDF. (b). The cumulative density function (CDF), F X (x).

8 vii 8. Continuous Random Variables: Find the value of the question mark within the box. You may use drawings for ease of understanding. (10 points). (a) Gaussian PDF (b) Exponential PDF? 1 2πσ e 1 2( x µ σ ) 2 dx = 0.5? λe λx dx = 0.5

9 viii Formulae: 1. Properties of Probabilities: P r (A B) = P r (A) + P r (B) P r (A B) (1) = P r (A) + P r (B) P r (A)P r (B), if A and B are independent events (2) = P r (A) + P r (B), if A and B are mutually exclusive events (3) 2. Conditional Probability P r (A B) = P r(a B) P r (B) (4) 3. Bayes Theorem P r (A k B) = P r (A k ) P r (B A k ) n i=1 P r(a i ) P r (B A i ) (5) P r (B) = n P r (B A k )P r (A k ) (6) k=1 4. Bernoulli Distribution ( n x P X (x) = ( ) n = x ) p x q n x, x = Number of successes in n trials (7) n! x!(n x)! 5. Multinomial Distribution P X1,X 2,...,X k (x 1,x 2,...,x k ) = n! x 1!x 2! x k! px 1 1 px 2 2 px k k, 0 x i n, (8) k and x i = n (9) i=1 x i = Number of successes of the ith kind (10) 6. Geometric Distribution P X (x) = pq x 1, 0 x < (11) x = Number of trials before a success occurs (12) 7. Cumulative Density Function F X (x) = x 0 f X (x) dx (13)