ece603_midterm2_2002

ece603_midterm2_2002 - ECE 603 - Probability and Random...

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Unformatted text preview: ECE 603 - Probability and Random Processes, Fall 2002 Midterm Exam #2 November 20th, 6:00-8:00pm, Marston 132 Overview The exam consists of six problems for 120 points. The points for each part of each problem are given in brackets - you should spend your two hours accordingly. The exam is closed book, but you are allowed two page-sides of notes. Calculators are not allowed. I will provide all necessary blank paper. Testmanship Full credit will be given only to fully justified answers. Giving the steps along the way to the answer will not only earn full credit but also maximize the partial credit should you stumble or get stuck. If you get stuck, attempt to neatly define your approach to the problem and why you are stuck. If part of a problem depends on a previous part that you are unable to solve, explain the method for doing the current part, and, if possible, give the answer in terms of the quantities of the previous part that you are unable to obtain. Start each problem on a new page. Not only will this facilitate grading but also make it easier for you to jump back and forth between problems. If you get to the end of the problem and realize that your answer must be wrong, be sure to write “this must be wrong because . . . ” so that I will know you recognized such a fact. Academic dishonesty will be dealt with harshly - the minimum penalty will be an “F” for the course. 1. [10] A point is chosen at random within a circle of radius 3 (i.e. all points within the circle are equally likely). The outcome (or observation) for the experiment is the distance of the point from the center of the circle. Define a non-trivial probability space for this experiment; that is, find , where is the observation space, is a set of subsets of to which probabilities are assigned, and is a probability mapping from to .  ¨£ ¥£ ¡ ©§¦¤¢ ¡  £ ¥ ¡ ¥   ¨ 2. A fair six-sided die is rolled once. Let be the number of spots appearing on the top of the die. A coin is then flipped times, and the number of heads (call it ) is recorded.    . , find the probability that % "$  ! #  [7] (b) Given that ! " [8] (a) Find the probability that . 3. Suppose that your goal is to maximize the profit of your business. If you decide to travel to Xville, the profit (in dollars) for your business is a random as given below. If variable with cumulative distribution function you decide to travel to Yville, the profit (in dollars) for the trip is a random variable with cumulative distribution function as given below. The random variables and are independent. 0 21 ( )' 6 87 & 4 5' 3 3 6 B¢ & 0 21 4 A' ( 9' 1 1 @ @ @ 2 3 0 1 1 2 3 [8] (a) To which city (Xville or Yville) do you travel to maximize the profit for your company. Be sure to justify your answer. [7] (b) Suppose you have some extra time so you travel to both Xville and Yville. Let the random variable be your total profit from the two trips. Sketch the cumulative distribution function for the random variable .  GE I C H¨ E GF C D 5' E  ¨£ ¥£ ¡ ©§¦¤F T VUH¨ 4. [20] An experiment is defined by the probability space , is the Borel -algebra restricted to , and , where is defined 6 @ -1  R£   S ¥  R£  QP ¡ ¤ ¥¢ ¡  ¢ ££ ¡ H¨ by . A sequence of random variables is defined as follows. for all . If is an irrational number, If is a rational number, for all . Does this sequence of random variables converge? If so, in what ways and to what limiting random variable does it converge? ¦  ©  ¦ § ¨&  © § flip %$  & & © ! 3  . . © )  © ! 3 3 .  § &  &  " © ! 3 , the autocorrelation function of     , the mean function of  " # ¦ ¦  © ! £ © 4 §' ' ) ( 4 [10] (c) Find . If the  & §  [5] (a) Find the probability mass function of [5] (b) Find  flip is “heads”, I let .     5. I flip a coin repeatedly. If the is “tails”, I let . Let [10] (d) Does the sequence of random variables converge? If so, in what ways and to what limiting random variable does it converge? [Be sure to justify your answer]. © ) 3 0 1   0 1 and be zero-mean, wide-sense stationary Gaussian random 6. Let processes that are independent of one another. (Recall that a Gaussian random process is one for which any collection of samples is jointly Gaussian). Assume that the two processes have the same autocorrelation function . (Note that ). Define the random process as:  0  ( ( R S R `X P 0 Ucb aYH 1    0 3 C 0 1 V W 0 3 C 0 21 . . F9 f D 2 3 ( ( C 0 1  &   ( 4 0 1  RS UTR [email protected] 3 PIG 0  2 3  DC EA F D C A9 86 4 £EB@!75! . , the autocorrelation function of  0 1 d0 eY£ C 0 3 % ' D D ¤( [10] (c) Find the probability density function  & [8] (b) Find , the mean function of ! 3 [5] (a) Find d e0 0 d e0 0 [7] (d) Pick any two times and , , that you want (your choice!) and give the joint probability density function of the random variables and . (For example, you might choose and ; then, you just have to give ).   5% 0 d 50 £ % 0 1  50 £ d F  9 1F i 9 ph Df D g % %  % 0 7 D Fr pFq es9 1eB9 D f d0 e3 C  % 0 3 C ! # d e0 ...
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This note was uploaded on 09/16/2011 for the course ECE ECE603 taught by Professor Dennisgoeckel during the Fall '10 term at UMass (Amherst).

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