ECEN 5612, Fall 2007 Noise and Random Processes Prof. Timothy X Brown NAME: CUID:

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1 Midterm ECE ECEN 562, Fall 2007 Noise and Random Processes Prof. Timothy X Brown October 23 CU Boulder NAME: CUID: You have 20 minutes to complete this test. Closed books and notes. No calculators. If you can not do a calculation by hand, set up the calculation and I will compute it while grading. You may use a single sheet of paper with any notes that you wish. You must turn in this sheet with your exam or you will lose points. For computational questions circle your answer. For multiple-choice questions, if you have trouble choosing, put your best answer with an explanation. If you think that there is not enough information to answer a question, make an assumption, write it down, and continue working. Show your work where possible; I cannot give partial credit otherwise. All problems are worth 0 points. This test, like all your activities at the University is subject to the honor code.

2 Helpful Information Distribution P X (x) or f X (x) E[X] Var[X] θ X (t) p x = 0 Bernoulli p x = p p( p) p + pe t ( ) n p Binomial x x ( p) n x x = 0,, 2,... n np np( p) ( p + pe t ) n Geometric Poisson Discrete Uniform Exponential Gaussian Uniform { p( p) x x =, 2,... { α x e α x = 0,, 2,... x! { x = k, k +,..., l l k+ { λe λx x 0 σ e (x µ) 2 { 2π b a p p p 2 pe t ( p)e t α α e α(et ) k+l 2 λ (l k)(l k+2) 2 λ 2 e tk e t(l+) e t λ λ t 2σ 2 < x < µ σ 2 e tµ+t2 σ 2 /2 a < x < b x Q(x) Markov Bound P [X x] µ X x Chebyshev Inequality P [ X µ X c] σ2 X c 2 One-Sided Chebyshev Inequality P [X µ X c] σ2 X σx 2 +c2 Chernoff Bound P [X c] min t 0 e tc θ X (t) Gaussian Tail Approximation P [Z c] = Q(c) < 2πc e c2 2 Sterling s Formula n! 2πn(n/e) n a+b 2 (b a) 2 2 e at e bt t(a b) 2

3 . Consider the experiment of rolling a six-sided die. Valid σ-fields are (circle all that apply): (a) {{}, {2}, {3}, {4}, {5}, {6}} (b) {, {, 2, 3, 4, 5, 6}} (c) {{, 2}, {3, 4, 5, 6}} (d) {, {, 2}, {3, 4, 5, 6}, {, 2, 3, 4, 5, 6}} (e) {, {}, {2}, {, 2}, {3, 4, 5, 6}, {, 2, 3, 4, 5, 6}} 2. Given two classes ω and ω 2 defined on a continuous-valued feature X. Which of the following (taken one at a time) would reduce the classifier error to zero. (circle all that apply): (a) P [ω ] 0 (b) (E[X ω ] E[X ω 2 ]) 0 (c) E[X 2 ω ] 0 (d) (E[X 2 ω ] E[X 2 ω 2 ]) 0 (e) (E[X 2 ω ] + E[X 2 ω 2 ]) 0 3. Given an ordinary deck of 52 playing cards, you are dealt four cards and order does not matter. (circle all that apply) (a) The number of ways we can draw no Kings is 48! 4!44!. (b) The number of ways we can draw 2 Kings and 2 Queens is 4!4! 2 4. (c) Let n h, n d, n s, n c be the number of hearts, diamonds, spades, and clubs. The number of possible combinations of (n h, n d, n s, n c ) is 7! 4!3!. (d) The event no Kings is independent of the event no Queen s. (e) The event no Kings is independent of the event no red card. 4. Given an ordinary deck of 52 playing cards the value of the card (Ace, 2, 3,..., Queen, King) is numbered, 2,..., 2, 3. You are dealt two cards, the first has value X, and the second has value Y. (a) X and Y are positively correlated. (b) X and Y are uncorrelated. (c) X and Y are independent. (d) If the two cards are different color, X and Y are positively correlated. (e) If the two cards are different color, X and Y are uncorrelated. 3

4 5. X and Y are independent random variables with MGF θ X (s) = e 2s2 and θ Y (s) = e es. Z = X + Y. Var[Z] = 6. You are investigating a fire alarm for your house. The performance data for the alarm states that on a given day: the alarm sounds if there really is a fire with probability 0.99; and the alarm sounds with probability 0.0 if there is no fire. A typical house has some sort of fire on a given day with probability What is the probability that there really is a fire given the alarm sounds? 4

5 7. Let σ 2 be the variance of arbitrary random variable X. Use Jensen s inequality to prove σ Let X be a uniform random variable on the interval [, +]. If Y = e X compute f Y (y). 5

6 In the next three problems X is an exponential random variable with mean λ. 9. Compute E[X n ] 0. Compute E[ X2 X2 ] and V ar[ ] 2 2. Prove or disprove that ˆµ n = n Xi 2 i= is a consistent estimator for. 2n λ 6

7 The next three problems are based on the following. Young kids are throwing darts at a dart board. Define a coordinate system on the wall so that the dart board is a 2 unit by 2 unit square centered at the origin. Let (X, Y ) represent the horizontal and vertical location of where the dart lands on the wall. The density is f XY (x, y) = 2π e 2 (x2 +y 2). 2. Compute the probability, p, a dart lands on the dart board. 3. Given p from the previous problem, if the kids throw 8 darts, compute the probability that only lands on the dart board. 4. At the end of the day, the kids are tired and most of the darts fall off the wall after landing. Let a be the average number that land on the dart board and do not fall down, compute the probability that only lands on the dart board. 7

8 The next two problems are based on the following. Given X = (X, X 2, X 3 ) T. X N(µ, K) where µ = (, 2, 3) T and 5. Compute the probability X 3 > K = Let A be the whitening matrix for K, i.e. A T KA = I. Compute A T A. 8

9 In the remaining four problems you will get full credit for any constructive feedback provided. 7. List two things this class is doing well. (a) (b) 8. List two things this class could do better. (a) (b) 9. Comment on the text in terms of your ability to learn the material from it? 20. Which lecture was (a) the best so far? (b) the worst so far? 9