EE 278 October 26, 2017 Statistical Signal Processing Handout #12 EE278 Midterm Exam

Transcription

1 EE 278 October 26, 2017 Statistical Signal Processing Handout #12 EE278 Midterm Exam This is a take home exam. The total number of points is 100. The exam is open notes and open any electronic reading device, provided they are used solely for reading material already stored on them and not for any other form of communication or information retrieval. Calculators are permitted though not needed. You may cite any result you use from the lecture notes (no need to rederive it). Begin each problem on a new page. You are required to work on the exam on your own, NO collaboration and NO consulting anybody except the course staff. You are bound by the Stanford Honor Code in this regard. Please sign the honor code (provided on the next page) and submit it with your exam. Please scan it if you submit the exam online. We will not grade your exam without the signed honor code. Please do not share the question sheet with any one else for the next 7 days. The exam is due back to us BEFORE 2 p.m. Friday, October 27th. Please turn in your midterm with Kara Marquez at Packard 267 (for written/printed portions) and/or upload it on Canvas (for online portions). If you have any comments or problems with submission please Pin Pin (pinnaree@stanford.edu). We are not responsible for any illegibility due to the quality of electronic scanning. Please ensure the scans are of high enough quality. Good luck and have fun!

2 The Stanford University Honor Code 1. The Honor Code is an undertaking of the students, individually and collectively: (a) that they will not give or receive aid in examinations; that they will not give or receive unpermitted aid in class work, in the preparation of reports, or in any other work that is to be used by the instructor as the basis of grading; (b) that they will do their share and take an active part in seeing to it that others as well as themselves uphold the spirit and letter of the Honor Code. 2. The faculty on its part manifests its confidence in the honor of its students by refraining from proctoring examinations and from taking unusual and unreasonable precautions to prevent the forms of dishonesty mentioned above. The faculty will also avoid, as far as practicable, academic procedures that create temptations to violate the Honor Code. 3. While the faculty alone has the right and obligation to set academic requirements, the students and faculty will work together to establish optimal conditions for honorable academic work. I acknowledge and accept the Honor Code. (Signed) Page 2 of 8 EE 278, Autumn 2017

3 1. Short questions (27 points). a. (3 points) Assume that you are able to sample a random variable X from an exponential distribution with parameter 1 (f X (x) = e x for x 0). Explain how to generate: i. (1.5 points) a Bernoulli random variable Y with parameter 1. e 2 ii. (1.5 points) a random variable Z uniformly distributed between 1 and 3. b. (4 points) Suppose F (x) is the cumulative distribution function (CDF) of a continuous random variable X. What is the variance of Y = e F (X)? c. (2 points) You bump into a student in the Main Quad. If the average age of Stanford students is 20 years, bound the probability that the student you bumped into is over 30 years old. d. (3 points) Compare E[X Y 3 ] with E[X Y ]. (Choose,, = or not comparable ). e. (3 points) Compare the MSE in optimal estimation of X based on Y with the MSE in optimal estimation of X based on Y 2. f. (3 points) Let X and Z be uncorrelated random variables with zero mean and variance N. Set Y 1 = Z and Y 2 = XZ. What is the MMSE estimate of X given Y 1 and Y 2 and its MSE? g. (3 points) Suppose X is a random variable and M X (t) = E[e tx ] exists for all t R. For t < 0, compare P (X α) with e αt M X (t). h. (3 points) Σ is a n n matrix with entries Σ ij = min{i, j}. Prove this is a covariance matrix by describing a construction of an n dimensional random vector with covariance matrix Σ. Hint: Start with n iid zero mean, unit variance random variables. i. (3 points) Σ is a n n matrix with entries Σ ii = 1 and Σ ij = ρ for ρ (0, 1) and i j. Prove Σ is a covariance matrix by describing a construction of an n dimensional random vector with covariance matrix Σ. Hint: Start with n 1 iid zero mean, unit variance random variables. EE278 Midterm Exam Page 3 of 8

4 2. Online shopping (15 points). You are a data scientist at a large social network company which also sells virtual reality (VR) headsets. You want to predict whether a customer would buy the VR headset using data from your company s online sales portal. Your company shows a nice advertisement of the VR headset on the sales portal and asks customers if they want to purchase the headset. Past data shows that a customer decides to buy the headset with a 25% probability. You model this with a random variable B Bern(0.25), where B = 1 means that the customer buys the headset and B = 0 otherwise. Your company also offers a one week free trial program to customers who want to try out the VR headset. Data shows that customers who have bought the headset are 90% likely to have tried it, and, conversely, those who decided not to buy the headset are 70% likely not to have tried it. You model this with a random variable T {0, 1} which has conditional pmfs given by: { 0.9 if t = 1 p T B (t 1) = 0.1 otherwise, and p T B (t 0) = { 0.3 if t = otherwise. You use the above model to predict the shopping behavior of new customers. a. (3 points) How effective is the free trial program? That is, what is the chance that a new customer who has tried the VR headset will buy it? From reading about social influence you realize that people are more likely to buy the VR headset if their friends have bought them. So you go back to past data to understand social influence on VR headset purchases. Let F be the number of friends a typical past customer had when they made their purchasing decision. You assume that given a customer s decision, the number of friends who ve bought the headset is independent of whether the customer has tried out the headset; i.e., F and T are conditionally independent given B. The data reveals that p F B (f b) is as follows: p F B (f 0) p F B (f 1) f f b. (3 points) What is the chance that a customer who has 3 friends who ve bought the VR headset will NOT buy it? (Hint: you re looking for an appropriate conditional probability involving B and F.) Page 4 of 8 EE 278, Autumn 2017

5 c. (4 points) A customer has not tried the VR headset but he has 3 friends who ve bought it. What is the chance that the customer will buy the VR headset? d. (5 points) Your boss wishes to understand which has the stronger effect on a customer s purchasing decision having no friends or not trying the product. How will you quantitatively answer her question? EE278 Midterm Exam Page 5 of 8

6 3. Restaurant ratings (27 points). In the city of Foodtopia there are n restaurants. Each year, food critics gather to judge and rank each of these restaurants and report their scores in the Gourmet magazine. Let S i be the score of the i th restaurant this year and let S i = 1/i. You are hired to predict next year s score and rank of the restaurants. You look at the score data from past years and build the following model. Let S i represent the score of restaurant i next year, then Γ i = S i/s i, 1 i n, are i.i.d. random variables with exponential distribution of rate 1. a. (3 points) What is the distribution of the random variables S i, i = 1,..., n? b. (4 points) What is the expected value of the minimum score for the next year,i.e. E[min i (S i)]? c. (8 points) What is the probability that restaurant 1, the top-ranking restaurant this year, will get the least score next year? d. (12 points) What is the probability that restaurant 1 will get the second least score next year? Page 6 of 8 EE 278, Autumn 2017

7 4. Time synchronization (31 points). Each morning the Timekeeper at a large Swiss clock company determines the offset of the company s master clock relative to GMT. The Timekeeper communicates the offset through a channel to other major clocks in the company. The channel has various additive noise values corrupting the offsets being transmitted. The channel is modeled as a directed graph with additive noise on the edges. As shown in Figure 1, let X 0 be the value of the offset the Timekeeper determines for the master clock, and let X 1, X 2,..., X 7, X 8 be the offsets communicated to the other clocks. If node i has a single parent in the graph, it receives X i = X parent Z i. If node j has multiple parents, it receives the following average value of its parents transmissions: X j = (X left parent Z j ) (X right parent Z j 6). 2 Model X 0 as a random variable distributed as N (0, P ). The Z i s, Z js and X 0 are all mutually independent. For i = 1,..., 8, Z i N (0, σ 2 i ) and, for j = 1, 2, Z j N (0, σ 2 j). Z 1 X 0 Z 2 Z 3 X 1 Z 4 Z 5 X 2 Z 6 X 3 X 4 X 5 X 6 Z 7 Z 1 Z 8 Z X 7 X 8 Figure 1: An illustration of the communication graph. a. (4 points) To determine the quality of the clock synchronization, suppose that the vector Y = [X 7 X 8 ] T is observed each day and the MMSE estimate of X 0 given Y is computed. Express the observation vector as a linear combination of X 0 and the noise vector, i.e. find matrix A such that Y = AX where X = [Z 1 Z 2... Z 6 Z 7 Z 1 Z 8 Z 2 X 0 ] T. b. (6 points) Assume that σ 2 i = 1 for i = 1,..., 8 and σ 2 j = 1 for j = 1, 2. Determine the MMSE estimate of X 0 given Y. Compute the corresponding MSE. Your answers should be in terms of P. EE278 Midterm Exam Page 7 of 8

8 For parts (c) and (d) suppose that it is required that 8 σi 2 i=1 2 j=1 σ 2 j = 10. (1) c. (9 points) Subject to the above constraint the Timekeeper may choose to have arbitrary values for σi 2 and σ 2 j. In other words, the Timekeeper is allowed to make some channels better by moving the noise power to other channels and making the latter channels worse. What are the best values of σ 2 i for i = 1,..., 8 and σ 2 j for j = 1, 2 so as to obtain the MMSE estimate of X 0 given Y? What is the corresponding MSE? d. (12 points) A mini-max solution. Suppose the Timekeeper wants to obtain the estimates ˆX 0,7 = E(X 0 X 7 ) and ˆX 0,8 = E(X 0 X 8 ) separately. You would have noticed that the solution in part (c) makes one of these two estimates really good while sacrificing the other. The Timekeeper deems this is not acceptable and seeks the mini-max solution: arg min {σ 2 i, i=1,...,8; σ 2 j, j=1,2} max {E(X 0 ˆX 0,7 ) 2, E(X 0 ˆX } 0,8 ) 2. Subject to the constraint at equation (1), what choice of values for σi 2 for i = 1,..., 8 and σ 2 j for j = 1, 2 yields the mini-max solution? What is the corresponding value of max {E(X 0 ˆX 0,7 ) 2, E(X 0 ˆX } 0,8 ) 2? Page 8 of 8 EE 278, Autumn 2017