ECE 650 1/11. Homework Sets 1-3

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1 ECE 650 1/11 Note to self: replace # 12, # 15 Homework Sets 1-3 HW Set 1: Review Assignment from Basic Probability 1. Suppose that the duration in minutes of a long-distance phone call is exponentially distributed, with probability density function: f(x) =.2 exp(-x/5), x > 0. Find the probability that the duration of a phone call a. be less than 2 minutes. Ans:.3297 b. will be between 5 and 6 minutes. Ans:.0667 c. will be less than 6 minutes given that it was greater than 3 minutes. Ans: A machine produces bolts the length of which (in cm.) is normally distributed with mean 5 and standard deviation = 0.3. A bolt is called defective if its length falls outside the interval (4.8, 5.2). a. What is the proportion of defective bolts produced? Ans:.5050 b. What is the probability that among 10 bolts produced by this machine, none will be defective? Ans: e-4 3. Three production lines are competing to see which line or team can finish assembling 100 phones the fastest. Their chances of finishing all 100 in less than 2 hours are 1/3, 1/5, and 1/12 respectively. Their production times are independent of each other. What is the probability that at least one team finishes all 100 phones in less than two hours? Ans: 23/45 4. A missile is known to hit its target with probability p. Given that in 5 trials it succeeded 3 times, what is the probability that it succeeded in the first trial? Hint: Try Bayes Theorem. Ans: 3/5 5. A fair coin is tossed (a) 3 times (b) 5 times. For case (a) and (b), find the probability that the number of heads is odd. 6. John and Harry are competing in an archery dual, with each trying to kill the other. First John shoots at Harry, then (if John missed) Harry shoots at John, etc. Both John and Harry are poor shots, with a probability of hitting their targets equal to 0.2. (Assume that any hits are fatal.) Find the probability that John will survive this ordeal i.e., find the probability that eventually John will kill Harry. Hint: you will probably need the formula for the sum of an infinite geometric series. 7. (Papoulis, 2.14) Consider two mutually exclusive sets, A and B, that are events for some experiment. Can they be independent? Why or why not?

2 ECE 650 2/11 8. (Papoulis, 2.18) Box 1 contains 1000 bulbs, 10% of which are defective. Box 2 contains 2000 bulbs, 5% of which are defective. Two bulbs are selected from a randomly selected box. a. Find the probability that both are defective. Ans: b. If both selected bulbs are defective, find the prob. that they came from Box (Gubner) A photo-sensor fails to activate if it receives fewer than four photons per ms. The number of photons per ms is a Poisson RV with parameter = 3. Find the probability that the sensor activates. Ans: Find the mean and variance of a RV that is uniformly distributed on the interval (0, 4). (Proof or derivation not required if you know the answer/shortcut.) 11a. Use MATLAB to verify your answers for problems 1, 2, and 9. Hint: Use functions of the form distcdf, where dist is norm, exp or poiss. (Also see MATLAB Addendum for Lecture 1 posted on the course web page. 11b. Use MATLAB to plot the pdf s for problems 1, 2, and 9. Hint: Use functions of the form distpdf, where dist is norm, exp or poiss, to generate the pdf s. Use MATLAB s subplot function to plot all three distributions on a single page. Also see the MATLAB Addendum described in part (a). 12. If X is uniformly distributed between 2 and 10, find (a) Pr(3 < X < 7); (b) Pr(X > 8). 13. Use MATLAB to: simulate the experiment of tossing a fair die 1,000 times. construct a 6-bin density histogram for the results of the experiment. Hint: To generate the variates, use a function of the form distrnd, where dist is unid. (Also see MATLAB Addendum for Lecture 1 posted on the course web page.) 14. Consider Examples 2.9 and 2.10 from the textbook (Miller & Childers), in which MATLAB code is written to simulate the tossing of two dice, and finding the probability that the sum is 5. Modify this code using the function unidrnd to simulate the toss of the dice. Run your modified code to simulate 10,000 tosses of the dice, and find the resulting approximation for Pr(sum = 5). 15. In a binarycommunications system, transmitted 0 s are incorrectly received as 1 s with probability.01, while transmitted 1 s are incorrectly received as 0 s with probability.02; find the total probability of error if 40% of the transmitted symbols are 0 s. 16. M&C, Prob. 2.64a. Ans. for 16:.0171

3 ECE 650 3/ Suppose that we start with a standard normal RV X ~ N(0, 1). Now transform X to form the new RV Y = 3X + 7. Write the equation for the pdf for the RV Y. (Proof or derivation not required if you know the answer/shortcut.) End of HW Set 1 HW Set 2 From Lecture 1: 1. Show that: a Gamma RV with parameter c = 1 is an exponential RV. 2. On a single sheet of paper, use MATLAB to graph four cases of the gamma pdf, with b and c values: (b, c) = { (2, 2), (4, 2), (2, 4), (4, 4) }. 3. Miller & Childers, Problem Ans. c: Miller & Childers, Problem 3.35b. (Assume 2 = 1 for the sake of the plot.) 5. Miller & Childers, Problem 3.36a. (Hint: See Lecture 1, p. 50, and let A be the event: M = Miller & Childers, Problem 3.39a. 7. (Dougherty) On the average, a gram of radium emits 3.57 * alpha particles per second. The Poisson RV X counts the number of emissions observed per nanosecond. a. Find the rate for the RV X; i.e., the average number of particles emitted per nanosecond. (See Lecture 1, p. 25) b. Find the probability that exactly 30 particles are emitted in a single nanosecond. c. Use MATLAB to verify your answer. Recall that for discrete RV s, the pdf, f X (x), or pmf actually gives the probability for any value x. 8. (Stark & Woods) A switchboard receives on the average 16 calls per minute. If the switchboard can handle at most 24 calls per minute, what is the probability that in any one minute the switchboard will saturate. Ans: Let X be a Gaussian RV, N(, x 2 = 3). Let W = 10 x/10, so that W db = 10 log 10 (W) = X. Find the equation for the pdf for RV W. Hint: Compare to Lecture 1 s discussion on lognormal RV s.)

4 ECE 650 4/ (Dougherty) The time to failure (in yrs) of a radar system is gamma distributed with b = ½ and c = 1.5. What is the probability that the system functions for at least one year prior to failure. Ans: A temperature monitoring system for a computer room is designed to sound a warning (and activate the air conditioning) whenever the temperature rises above 75 F, which happens 20% of the time. If the temp. is > 75 F: the voltage X is N(4, 2 = 4): X = signal + noise If the temp. is 75 F: the voltage X is N(0, 2 = 4): X = noise only Find the over-all pdf for the RV X representing the system voltage. Use MATLAB to plot the over-all pdf. 12. A Gaussian RV X is N(1, 2 = 2). Find the conditional pdf: f X X>2 (x); i.e., find the conditional probability density function for X, given that X > 2. Use MATLAB to sketch both the original pdf and the conditional pdf on the same plot. From Lecture 2: 13. Find (a) the moment generating function, and (b) the characteristic function, for the random variable X that is uniformly distributed on [-4, 4]. 14. Find (a) the moment generating function, and (b) the characteristic function, for the random variable X with PMF as shown: P X (x) (1/2) (1/2) x 15. Use the moment generating functions from answers 13 and 14 to find the first moment for RV X in problem 13 and both the first and second moments for the random variables X in problem Find the variance for the random variables as given in problems 13 and A normal RV X: N(1, = 2) is input to a half-wave rectifier, with input/output characteristic: Y = X, X 0 0, X < 0 Find the pdf of the output RV, Y.

5 ECE 650 5/ (Papoulis, 5-6) Let X be uniform on the interval (0, 1). Find the density of the RV Y = - ln(x). 19. Assume that humans have an average height of 65 inches, with a standard deviation of 3 inches. Also assume that the distribution is symmetric about the mean. Use the Chebyshev bound to find an upper bound on the probability that a human has height greater than 70 inches. Ans: 9/ Let X be uniformly distributed between 0 and 50. (a) Find the Markov bound on the probability that X is greater than or equal to 10. (b) Find the Markov bound on the probability that X is greater than or equal to 40. (c) Compare your answers from part (b) to the actual probability that X is greater than or equal to (Gubner) A cellular company knows that the expected (or average) number of simultaneous calls coming into a base station is 100. Since the actual number is a random variable, the station is designed to handle a maximum of 150 calls. a. Use the Markov inequality to find an upper bound on the probability that the station receives more than the maximum allowable number. Ans:.6623 b. If the variance in the number of incoming calls is 50, find a tighter bound on the probability that the station receives more than the maximum allowable number. xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx HW Set 3 1. (Dwight Mix) In shooting at a circular target, we assume that the points of impact in both the horizontal and vertical directions are independent Gaussian RV s, N(0, 2 ), where the bullseye is centered at the origin. For a particular shooter, the aimpoint variance is 2 = 4 in 2. (a) Find his average distance from the bullseye. 1 (b) Find the probability that a particular shot (by this same shooter) is more than 4 inches from the bullseye. Ans. b: a. Let X, Y and Z be independent, identically distributed (iid) RV s, each U(0, 2). Let W = X + Y + Z. Find fw(w). (This problem is for A students only!) 2b. Use MATLAB to simulate the underlying experiment for problem 2a. Specifically, generate 3 vectors X, Y, and Z, each containing 10,000 variates from the distribution U(0, 2). Then form the vector W = X + Y + Z, and make a (density) histogram of W. Plot 1 Once you know what type of rv you have, feel free to look up the equation for the average. You don t need to derive it! Similarly for the equation for the cdf if you do part b by hand instead of by MATLAB.

6 ECE 650 6/11 your theoretical answer (obtained in part a) on the same graph as your density histogram, and hope that they show good agreement! 3. Consider the experiment of tossing a single coin, with random variables X 1 and X 2 as shown: H T X X (a) Find and graph the joint pdf, fx1x2(x 1, x 2 ). (b) Find and graph the marginal pdf s, fx1(x 1 ) and fx2(x 2 ). (c) Find fz(z) if Z = X 1 + X 2. (Caution: X 1 and X 2 are not independent.) 4. (Papoulis 6-1a,f) Let X and Y be independent, identically distributed RV s: fx(x) = e -x U(x), fy(y) = e -y U(y) a. Find fz(z) if Z = X + Y. b. Find fz(z) if Z = max(x, Y). 5. Given the joint density fxy(x, y) = (1/8) e -(x/2 + y/4) U(x) U(y), a. Find f(x). b. Find f(y). c. Find f X Y (x). d. Find f Y X (y). e. Are X and Y independent? Why or why not? 6. Let X and Y be standard normal independent rv s. Find the probability that a random point (X, Y) falls in the shaded half-plane. Hint: it may be helpful to form a new RV, Z, based on the equation of the line dividing the plane. Ans:.1855 y 1 2 x 7. (Carol Ash) If RV X has mean 12 and variance 9, find a. var(10x) b. var(10x + 3) c. var(-x) d. E(X 2 ) 8. (Carol Ash) Find the variance of Z = X Y as a function of var(x), var(y) and cov(x, Y).

7 ECE 650 7/11 9. (Yates & Goodman, Ch. 9) The following table gives the joint probabilities for RV s X and Y: Pr(x,y) Y = -3 Y = -1 Y = 1 Y = 3 X = -1 1/6 1/8 1/24 0 X = 0 1/12 1/12 1/12 1/12 X = 1 0 1/24 1/8 1/6 a. Find the marginal pdf s f(x) and f(y). b. Are X and Y independent? 10. (Yates & Goodman, Ch. 5) Random variables X1, X2, X3, and X4 have joint pdf f(x 1, x 2, x 3, x 4 ) = k 0 < x 1 < x 2 < 1, 0 < x 3 < x 4 < 1 0 else First find constant k to make the joint pdf legitimate; then find the marginal pdfs f(x 2, x 3 ) and f(x 3 ). 11. (Papoulis, 4 th ed., 6.58) RV s X and Y are jointly distributed over the region 0 < x < y < 1, with: fxy(x, y) = kx 0 < x < y < 1 0 else Determine k to make the pdf legitimate, and find the covariance between X and Y. Ans: k = 6, cov = 1/ (Papoulis, 4 th ed., 6-71) RV s X and Y are uniform on the interval [-1, 1] and independent. Find the conditional density fr(r M) for the RV R = sqrt(x 2 + Y 2 ), where M = {R< 1}. Hint: x, y uniform on 2 x 2 square implies f(x, y) =, on the square. 13. (old quiz question) Let f X,Y (x,y) be the density function representing a jointly Gaussian pair of random variables, each having mean 0 and with covariance matrix 3 C 1 Write out f X,Y (x,y) explicitly (old quiz question) Find the correlation matrix for the random vector [X 1 X 2 ] T if X 1 and X 2 are independent, identically distributed RV s, each U(0, 12).

8 ECE 650 8/ (old quiz question) A telemetry signal, T, transmitted from a temperature sensor on a communications satellite is N(0, 9), where the second parameter is the variance. The receiver at mission control receives R = T + X where X is a noise voltage U(-3,3), independent of T. The receiver uses R to calculate linear MMSE estimate of the telemetry voltage: ^ T = AR + B a. Find the variance of RV R. b. Find the covariance of RV s R and T. ^ ^ c. Find the linear MMSE estimate, T = AR + B. Ans: T = (3/4) R 16. The input signal Y to a channel is N(0, 1), while the output (observable) is X = Y + N, where noise N is N(0, 2 = 9). (Assume that signal Y and noise N are independent.) a. Find the linear MMSE estimate of Y in terms of X; i.e., find A and B for the estimate Y = A X + B. b. Find the best (arbitrary) MMSE estimate of Y in terms of X; i.e. find the function c(x) that minimizes the mean squared error. 17. Use the technique for transformations of pairs of RV s (where input RV s X, Y yield output RV s W, Z) to find the equation for the pdf of RV Z = X Y, if the pdf s for X and Y are known and X and Y are independent. Hint: Let Z = XY; let W = Y; then find f ZW (z, w); then integrate to find fz(z). 18. From the handout on Conditional Probability and Estimation, Example 2, do parts c and d. (Note that the answers, but not the derivations, are given on the handout.) 19. Consider the 3-dimensional random vector : X = X 1 X2 X3 the Y k are independent Gaussian RV s with mean k and unit variance., where X k = k Y k, and where a. What is the mean vector, X? b. What is the correlation matrix, RX? c. Write an explicit expression (not in vector/matrix form) for the joint pdf f(x 1, x 2 ) for random variables X 1 and X 2. d. Write the correlation matrix, R, for random variables x 1 and x Let X be a R vector with mean X = and covariance matrix CX =

9 ECE 650 9/11 a. Find the correlation matrix, RX. b. Find the correlation, the covariance, and the correlation coefficient of random variables X 1 and X 2. Also find the variance of X 1 and the variance of X 2. c. Let W be a 2-dimensional, elementary white random vector. Design a linear transformation (X = HW + C) which operates on the elementary white vector to generate the R vector X described above. (In other words, find the required H and C for the transformation.) 21. (old test question) Two Gaussian random variables x 1 and x 2 have 0 means and variances 4 and 9 respectively. Their covariance Cx 1 x 2 is equal to 3. a. Find the covariance matrix, CX, for the random vector X = X 1. X2 b. Suppose that we generate two new random variables, Y 1 and Y 2, using the transformation H shown below, according to Y = HX: 1 2 H 3 4 Find the covariance matrix for random vector Y = Y 1 Y2 c. Write the joint probability density function, f(y 1, y 2 ), for the random variables Y 1 and Y (mostly from Mix) The following statements are either correct, partially correct, or wrong. Label the statements correct as is, or modify the statements as needed to make them correct. a. If events A and B are independent, then Pr(A B) = 0. b. If 2 RV s are independent, then they are uncorrelated. c. RV s X and Y are uncorrelated if E(XY) = 0. d. The conditional cdf F X Y (x) = F(x y) is the derivative (with respect to x) of the conditional pdf f X Y (x) = f(x y). e. The pdf for the n-dimensional Gaussian random vector can be written down if we know the means, the variances, and the covariances. f. The central limit theorem says that the sum of a large number of RV s is Gaussian, even though none of the RV s may be Gaussian. g. To apply Chebyshev s inequality to a RV X, we need to know the mean and variance of X. h. The linear transformation of a Gaussian RV is Gaussian. i. The sum of n Gaussian RV s is Gaussian.

10 ECE /11 j. If 2 RV s are uncorrelated, then they are independent. 23. A random binary signal, S, is equally likely to take values ±10. The signal into the receiver is R = S + N, where N is N(0, 2 = 9), and S is independent of N. Use MATLAB to plot f S (s), f N (n), and f R (r), one beneath the other on a single page. 24. (old test question, closed-book, closed-notes section, 5 pts) Find the covariance matrix for the white vector W = [W 1 W 2 W 3 ] T if the W 1, W 2 and W 3 all have standard deviation (old test question, closed-book, closed-notes section, 5 pts) Given Y = X + N, where X is uniform on (0, 10) and N is Gaussian, N(2, 2 = 9), and X and N are independent, find the correlation between random variables X and Y. 26. (old test question, 15 pts) Let X be uniformly distributed on (-2, 2). Let Y = X 2. Find the linear MMSE estimate ( Ŷ = AX + B) of Y in terms of X. 27. (old test question, hard) Consider a random variable Y, to be estimated: Y: U(0, 1). The scalar observation is X = Y + N, where Y and N are independent, and where f N (n) is: f N (n) a. (10 pts) Find f X (x) for the case: 2 < x < 3. (Hint: since X = Y + N where Y and N are independent, this is a convolution problem; however, you only need to find the answer for one case of shift value.) b. (10 pts) Find (ii) f X (x y). (Hint: sketch: f X (x y), and first write the equations for the case f X (x Y =.5); note that the only possible y values are between 0 and 1.) c. (10 pts) Use Bayes Rule for pdfs, together with your answers from parts a and b, to find f Y (y x) for the case: X = 2.5. n d. What is the best (in the MMSE sense) estimate of Y, given that X = 2.5? (Note: your answer should be a number, not a function.)

11 ECE / (Mix, and former test question) Two RV s X and Y are associated with the toss of a single fair die, taking values as shown: # on die RV Y RV X a. Find the constant c that is the MMSE estimate of the RV Y, and the resulting meansquared error; i.e., find Ŷ c, and find E[(Y-c) 2 ]. (Ignore RV X for part a.) b. Now find the MMSE estimate of the RV Y, as an arbitrary function of X (treating X as the observable ). Hints i iv below: You need to do this for each possible value of X. Also find the resulting MSE using your estimate. i. If X = 4, find Ŷ X 4 ii. If X = 1, find Ŷ X 1 iii. If X = 0, find Ŷ X 0 iv. If X = 9, find Ŷ X 9 Summary of Estimate: Ŷ if X = Ŷ if X = MSE of the estimate: