EE 351K Probability, Statistics and Stochastic Processes - Spring 2016 Homework 11 Topics: Point estimation, Condence intervals, Random Processes Homework and Exam Grading \"Philosophy\" Answering a question is not just about getting the right answer, but also about communicating how you got there. This means that you should carefully dene your model and notation, and provide a step-by-step explanation on how you got to your answer. Communicating clearly also means your homework should be neat. Take pride in your work it will be appreciated, and you will get practice thinking clearly and communicating your approach and/or ideas. To encourage you to be neat, if you homework or exam solutions are sloppy you may NOT get full credit even if your answer is correct. Q. 1. A sample of size n is to be taken to determine the percentage of the population planning to vote for the incumbent in an election. Let Xk = 1 if the kth person sampled plans to vote for the incumbent and Xk = 0 otherwise. We assume that X1 , . . . , Xn are i.i.d., and such that P(Xi = 1) = p. We also assume that p is close enough to 1/2 so that = p(1 p) 1/2. Let Sn = X1 + + Xn . 1. Suppose n = 900. Estimate the probability that Sn n p 0.025. 2. Suppose n = 900. Estimate the smallest number c such that P 3. Estimate the smallest required sample size n such that P Sn n Sn n p c 0.01. p 0.025 0.01. Q. 2. Joe's Trucking provides shipping service from the outlying areas of New York City to a known distribution point within Manhattan. To understand the number of trucks required for a given service level agreement dened by a guaranteed delivery interval, the operations manager conducts a study of drive times. The manager would like to obtain an accuracy of 10 min, plus or minus 5 minutes. Randomly sampling the drive times over a 30-day period, including weekends and all times of day, produces 300 data points. A 95 percent condence interval is required to ensure customer satisfaction. The mean is found to be 3.01 hours, with a standard deviation of 0.64. Is the sample size adequate for the accuracy requirements? If yes, determine the condence interval for the mean delivery time. Q. 3. A source emits a random number of photons K each time that it is triggered. We assume that the PMF of K is pK (k; ) = c()ek/ , k = 0, 1, 2, 3... where is the temperature of the source and c() is a normalization factor. We also assume that the photon emissions each time that the source is triggered are independent. We want to estimate the temperature of the source by triggering it repeatedly and counting the number of emitted photons. 1. Determine the normalization factor c(). 2. Find the expected value and the variance of the number K of photons emitted if the source is triggered once. 3. Derive the ML estimator for the temperature , based on K1 ...., Kn , the numbers of photons emitted when the source is triggered n times. 4. Show that the ML estimator is consistent. Q. 4. Shazam is a mobile phone app that helps users determine a song that may be currently playing whereever they are. Lets try to understand the basic working principle of the application. To keep things simple suppose the following 1 there are only two possible songs and they are equally likely the application records a few seconds of the song, which we will assume, again for simplicty, correspond to the rst few seconds of the song, giving a vector of length n samples. These are represented by two equal length vectors a = (a1 , a2 , ....an ) or b = (b1 , b2 , ....bn ) assume that the vectors have equal energy i.e., a2 + a2 + ....a2 = b2 + b2 + ....b2 . and assume that they are roughly orthogonal n n 1 2 1 2 i.e., < a, b >= . if song a is playing the recording is modelled by a vector X corresponding to the samples a plus IID Gaussian noise with mean zero and variance 2 , i.e., X = a + N where Xi = ai + Ni and Ni N(0, 2 ). A similar model applies if song b is playing. A basic strategy in determing which song is playing is suggested to you. Given you record the vector X = x determine if < x, a >< x, b > if so it is song a otherwise it is b. This particular structure for deciding is called a matched lter as we match the received signal (x1 , x2 ...xn ) with each one of the possible songs by inner products and then select the one which gives the highest value. 1. Prove that this rule corresponds to the MAP decision rule. 2. In practice the snippet you record is not the rst few seconds of the song, but could be any snippet of a given length. How would you address this problem? Q. 5. Text p. 326, Problem 1. Q. 6. Text p. 326, Problem 3. Q. 7. Customers depart from a bookstore according to a Bernoulli process with parameter p = 0.15 (per minute). Each customer buys a book with probability 2/3, independent of everything else. 1. Find the distribution of the time until the rst sale of a book. 2. Find the probability that no books are sold between 1:00-4:00 pm on some particular day. 3. Find the expected number of customers who buy a book between 1:00-4:00 pm on some particular day. 2