FINAL EXAM: Monday 8-10am

Transcription

1 ECE 30: Probabilistic Methods in Electrical and Computer Engineering Fall 016 Instructor: Prof. A. R. Reibman FINAL EXAM: Monday 8-10am Fall 016, TTh 3-4:15pm (December 1, 016) This is a closed book exam. There are 11 problems. Neither calculators nor help sheets are allowed. Some formulae and helpful information is attached at the end. Cheating will result in a zero on the exam and possibly failure of the class. Do not cheat! Use of any electronics is considered cheating. Put your name on every page of the exam and turn in everything when time is up. Name: PUID: I certify that I have neither given nor received unauthorized aid on this exam. Signature:

2 Problem 1. (Multiple choice: 5 points) Two fair dice one Red and one Blue are rolled. Let A be the event that the Red die shows an even number. Let B be the event that the Blue die shows an even number. Let C be the event that the sum of the numbers shown on the two dice is even. Which of the following is true? (If you show your work you may receive partial credit.) (a) Each pair of events is independent, but A, B, and C are not mutually independent. (b) A, B, and C are mutually independent. (c) Exactly one pair of the three events is independent. (d) Exactly two of the three pairs of events are independent. (e) There is no pair among the three events that is independent.

3 Problem. (5 points) The probability that a randomly selected tablet (like an ipad or a Microsoft Surface) has cellular connectivity is 0.5. A tablet with cellular is twice as likely to have a USB connector as a tablet that does not have cellular. Compute the probability that a tablet has cellular connectivity given that it has a USB connector. 3

4 Problem 3. (15 points) Let X be the number of bugs in the first draft of a piece of software, and let Y be the number of bugs remaining in the second draft of the same software. Their joint PMF is given by the following table: Y= /6 0 0 X= 1 1/3 1/6 0 1/1 1/1 1/6 (a) What is the conditional PMF of Y given X =? (b) What is the marginal PMF of X? (c) Find the Variance of X. 4

5 Problem 4. (15 points) You want to watch a 10-minute video. Due to the current network conditions, in each (discrete) minute of video watching, there is a chance p of having one rebuffering event, and probability 1 p of having none. A rebuffering event happens when there is not enough bandwidth to play the video continuously, and so the video playback stalls. The rebuffering events in each minute are independent of each other. (a) What is the PMF of the random variable X 10, which is the total number of rebuffering events experienced during the 10-minute video? (Note: You may assume that each playback stall is so short in time that it is negligible and can be ignored, when counting the length of the video.) Now, let X k be the discrete-time random process which counts the cumulative number of rebuffering events experienced up until the current time. For example, X 3 is the total number of rebuffering events that happened during minutes 1,, and 3, and X 10 is as described above. (b) Find the PMF for X k for k = 1,,..., 10. (c) What is the mean of the random process as a function of time k. 5

6 Problem 5. (15 points) Let X be a random variable with PDF f X (x) = c(x x ), for 0 x (a) Find c. (b) What is P (X < 1)? (c) Find E(X A) for the event A = {X < 1}. 6

7 Problem 6. (Multiple choice: 5 points) Let Y = X 1/3, and let X be a continuous random variable that is uniformly distributed on the interval [ 1, 8]. What is the PDF of Y? (Note: if you show your work you may get partial credit.) (a) (b) (c) (d) (e) f Y (y) = f Y (y) = f Y (y) = f Y (y) = f Y (y) = { 3y /513 for 1 < y < 8 0 otherwise { y /3 for 1 < y < 0 otherwise { 1/3 for 1 < y < 0 otherwise { 1/9 for 1 < y < 8 0 otherwise { 3y for 1 < y < 0 otherwise (f) None of the above. 7

8 Problem 7. (0 points) Let X and Y be random variables that have the joint pdf f X,Y (x, y) = c(x + y), for 0 x 1, 0 y 1 (a) Find c. (b) Find the marginal PDFs of both X and of Y. (c) What is P (X > Y X < 1/)? (d) Find f Y (y x). 8

9 Problem 8. (5 points) The number of likes to a social-media post on any given day is a Poisson random variable, with mean α. However, the parameter α is a random variable that depends on the amount of sunshine outside and is uniformly distributed on the continuous interval [0, 3]. What is the probability there is exactly one like on a specific day? (Hint: you may find the following integral useful.) ( x xe ax dx = a 1 ) a e ax 9

10 Problem 9. (5 points) Let X be a Gaussian random variable with mean 5 and variance 4. What is the probability that X > 57 given that X > 54? (You may leave the (approximate) answer in terms of a ratio of integers.) 10

11 Problem 10. (5 points) Let X, Y, and Z be random variables, where X and Y are uncorrelated. The means of the RVs are E(X) = 1, E(Y ) =, and E(Z) = 1, and E(XZ) = 5. What is COV (X, Y + Z)? 11

12 Problem 11. (Multiple choice: 5 points) Let X and Y be independent random variables, with moment generating functions and M X (s) = 0.8 exp(s) + 0. exp(s) M Y (x) = 0.5 exp(s) exp(s) respectively, for < s <. What is the E(Z ) where the random variable Z is defined by Z = X + Y? (If you show your work you may receive partial credit.) (a) 0.5 (b).7 (c) 4.1 (d) 7.7 (e) None of the above 1

13 Empty page to show more work. Label problems clearly! (I need to be able to find your work to give you credit.) 13

14 Discrete Random Variables Bernoulli Random Variable, parameter p S = {0, 1} p 0 = 1 p, p 1 = p; 0 p 1 E(X) = p; VAR(X) = p(1 p) M X (s) = 1 + p + p exp(s) Binomial Random Variable, parameters (n, p) S = {0, 1,..., n} p k = ( n k) p k (1 p) n k ; k = 0, 1,..., n; 0 p 1 E(X) = np; VAR(X) = np(1 p) M X (s) = (1 + p + p exp(s)) n Geometric Random Variable, parameter p S = {0, 1,...} p k = p(1 p) k 1 ; k = 0, 1,..., ; 0 p 1 E(X) = (1 p)/p; VAR(X) = (1 p)/p M X (s) = pe s /(1 (1 p)e s ) Poisson Random Variable, parameter α S = {0, 1,...} p k = α k e α /k! k = 0, 1,..., E(X) = α; VAR(X) = α M X (s) = exp(α(e s 1)) Uniform Random Variable S = {0, 1,..., L} p k = 1/L k = 0, 1,..., L E(X) = (L + 1)/; VAR(X) = (L 1)/1 M X (s) = (1 exp(s(l + 1)))/(1 exp(s)) Continuous Random Variables Uniform Random Variable Equally likely outcomes S = [a, b] f X (x) = 1/(b a), a x b E(X) = (a + b)/; VAR(X) = (b a) /1 M X (s) = (exp(bs) exp(as))/(s(b a)) Exponential Random Variable, parameter λ S = [0, ) f X (x) = λ exp( λx), x 0, λ > 0 E(X) = 1/λ; VAR(X) = 1/λ M X (s) = λ/(λ s) One Gaussian Random Variable, parameters µ, σ S = (, ) f X (x) = exp( (x µ) /(σ ))/ πσ E(X) = µ; VAR(X) = σ M X (s) = exp(µs + s σ /) 14

15 Two Joint Gaussian Random Variables, parameters m X, σ X and m Y, σ Y S X = (, ), S Y = (, ) 1 f X,Y (x, y) = πσ X σ Y 1 ρ XY [ ( (x ) 1 mx exp (1 ρ XY ) ρ XY σ X ( ) ( x mx y my σ X σ Y ) ( ) )] y my + σ Y E(X) = m X ; E(Y ) = m Y ; VAR(X) = σx VAR(Y ) = σy Trigonometric identities Other useful formulas exp(jθ) = cos θ + j sin θ sin θ = sin θ cos θ; cos θ = cos θ 1 cos A cos B = 1 (cos(a + B) + cos(a B)) sin A cos B = 1 (sin(a + B) + sin(a B)) cos A sin B = 1 (sin(a + B) sin(a B)) sin A sin B = 1 (cos(a B) cos(a + B)) k=0 n k=0 kr k 1 = k=1 r k = 1 rn+1 1 r r k = 1 1 r n k=1 n k=0 if r < 1 1 (1 r) if r < 1 n k = k=1 n(n + 1) k = n3 3 + n + n 6 k=0 x k k! = ex ( ) n a k b n k = (a + b) n k 15

16 Table 1: Table of the Standard Normal Cumulative Distribution Function Φ(z) z