ece603_midterm2_fall2010

Ece603_midterm2_fall - ECE 603 Probability and Random Processes Fall 2010 Midterm Exam#2 November 22nd 6:00-8:00pm Goessman 20 Overview The exam

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Unformatted text preview: ECE 603 - Probability and Random Processes, Fall 2010 Midterm Exam #2 November 22nd, 6:00-8:00pm, Goessman 20 Overview The exam consists of four problems for 130 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. You may find the following fact useful as you solve this exam: Fact: Given a collection of jointly Gaussian random variables, every linear combination is Gaussian. Conversely, given that every linear combination of a collection of random variables is Gaussian, that collection of random variables is jointly Gaussian. have joint probability density function: 3 )  ) 63 )  ) @91%8754210( and ¡ 1. The random variables  A( " #!©§ ¦¤¢    ¨ ¥ £   $  '&%¨ otherwise [10] (a) Find . "  ¨ £ DCB¢  G E I H¨ £ F§ ¢ , the conditional probability density function of given . For this, write between constants and your limits on as (potentially) functions of .  ¡   [5] (c) Find your limits for . , the marginal probability density function of [5] (b) Find [5] (d) Suppose that you are planning to estimate from an observation . Find the such that when , you are “best” able to estimate from . In other words, if you are trying to estimate and are hoping for an that makes it easy, what would you choose? ¡  ¡ (Note: Some of these answers are short): , let  W , let be ¡ ¡  U V¡ ¨ G S G T [5] (b) Let be a random variable uniformly distributed between 0 and 1. Given Gaussian with mean and variance . Find .  ! be a Gaussian random variable with mean 0 and variance 1. Given [7] (a) Let uniformly distributed between 0 and . Find . ¡ ¡  P  R and  Q 2. More multiple random variables be  U V¡ ¨ T  X be a random variable uniformly distributed between 0 and 1. Let let be Gaussian [8] (c) Let with mean and variance . Suppose and are independent. Find (you can leave an integral expression here). (restricted to h3 igA( e T ¡ q rpc igA( fd` h3 e  . ¨   U V¡ ¨ T T X s v a¨ 8 y €8  c ` 2ba¨ . Find U Y¡ 8 ( "   v s¨ xwgut¨ T   R‚x¨ ¡   €Rx¨ " Suppose ,  Find . . be given by ¡ [10] (d) Let the probability space course), and , of 3. Suppose that I am interested in characterizing two random variables and (both individually and their relation to each other). I know in advance that the variances of the two random variables are equal ( ), but I do not know . I know nothing about either mean, including whether they are equal or not. Unfortunately, I am not able to observe the random variables individually, but only and . Making a bunch of measurements, I come up with the following: 8 8  y  Ih £ ¤Xy . .  $ , and , respectively. Answer these © ¡  $ ¡ ¡¢ ¡¢ ¡¢ ¡ ¢ e e ¨e ¨e ¡ 8 $ £ [15] (c) You have some extra time on your hands, so you define and , the probability density functions and estimate three parts separately: and ¡ § x£ ¥ b¨ ¢  § , the correlation coefficient of ¡ ¡ ¡ h e ¡ h  8¡y h  $ ¥ ¦X e  Ih and y ¡ ¡y and . . h 8 8  ¡ . XX § [10] (b) Find 8 [5] (a) Find ¡ . ¨ © ¨ ©     x¢ ¨ ¨ Suppose that you note that is Gaussian and is not Gaussian. Are and jointly Gaussian? (possible answers are yes, no, or maybe - a short justification is fine.) Suppose that you note that is Gaussian and is Gaussian. Are and jointly Gaussian? (possible answers are yes, no, or maybe - a short justification is fine.) Suppose that you note that and are jointly Gaussian. Are and jointly Gaussian? (possible answers are yes, no, or maybe - be rigorous here with your justification.) ¡ ¡ ¡ , of course), . 8   © $ #Fx¨ ¨    X h A( e  ¨ © q  X c h A( e   4` ¨   c ` 2ba¨ 4. (a) Let the probability space be given by , (restricted to and . Let a sequence of random variables be defined by 8    T s    G 8 ¨  y v   v s¨ x g t¨ T #! $ E£ "£ ¢ [8] Find , the conditional probability density function of given . [12] Determine whether the sequence converges, and, if so, to what and in what ways? Consider almost sure convergence, mean square convergence, convergence in probability, and convergence in distribution. You can do them in any order that you want, and you can use one type of convergence to imply another when appropriate.  x¨  X X h i3 53  e y & ' % y  53 e  @`  be given by , (restricted to . Let a sequence of random variables be defined by (( ) ) q RWc i3 h (( )X )X ,  T s v   g t¨  v s¨ y  c ` 2ba¨ 1( 53 8 2)X  ‚  ‚ 3 4  ¨ 1( 2)X y (  R  ¨  D¨  T  #£ 0B¢ [5] Find , the probability density function of [10] Determine whether the sequence converges, and, if so, to what and in what ways? Consider almost sure convergence, mean square convergence, convergence in probability, and convergence in distribution. You can do them in any order that you want, and you can use one type of convergence to imply another when appropriate. & ' % (b) Let the probability space of course), and .  y X ¨ [10] (c) Let be integer random variables such that . . Determine whether the sequence converges, and, if so, to what and in what ways? Consider mean square convergence, convergence in probability, and convergence in distribution. You can do them in any order that you want, and you can use one type of convergence to imply another when appropriate. T X T & 9 % 1 887g X 666 3 ...
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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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