ece603_midterm2_2006

ece603_midterm2_2006 - ECE 603 - Probability and Random...

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Unformatted text preview: ECE 603 - Probability and Random Processes, Fall 2006 Midterm Exam #2 November 13th, 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-side 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. You are on your first job and are asked to characterize a random variable . By making a large number of observations, you form a normalized histogram (it integrates to 1) as shown below. Using a computer program, you fit the following function to your histogram (i.e. this function is your estimate of the probability density function of ): & % $     ¨ ¦¤ ¢ #!  " ©§¥£¡  13 54 1 2 ) 0( ¨ ¦¤ ¢ ©§¥'¡ ) 0( (i.e. be defined by: FI QP FD GE 7 , the cumulative density function (CDF) of . 7 B HF @8 A9 B C 7 ¨U V§¤ S TR Find ) [10] (b) Let the random variable , and Var 1 6 [10] (a) Use the estimated probability density function to find , the expected value of , the mean squared value of , and the variance of ). 0.25 0.2 f(x) 0.15 0.1 0.05 0 x 2 4 Figure 1: Normalized histogram of ) 1W YXB ¦ . Given , the marginal probability density function of b f7 , the probability that 8 10 for Problem 1. c egW c ¨ edW ¨U V§¤ b 97 S ¡ ¤ ) [12] (a) Find a [8] (b) Find is uniform on 1WB YXHF 2. Suppose that the random variable variable on . 6 . 7 −2 −4 . ¦ ` −6 , 7 −8 0 −10 is a uniform random 3. A coin is flipped three times. Let the random variable be the number of heads in the first two flips of the coin, and let the random variable be the number of tails in all three coin flips. 7 and . The easiest way to express your answer is probably a ¨ VU  7 , the covariance of and . 7 ¦ B¦ ¨ 7  U a ¤ B C ¤ ¦ [7] (b) Find cov for all U [8] (a) Find table in and . 4. The money (in thousands of dollars) made from investing in stocks “Ystock” and “Zstock” are modeled as the random variables and , respectively. Assume and are independent with respective probability density functions and as shown below: ¡ ¡ 7 ¨ ¤¤ ¢ '¥£¡ ¨ ¥£¡ ¤¤ ¢ ¨U V§¤ S 7 ¡ ¨U V§¤ § © S ¡ § © @ @ ¨ ¨ W ¤¦ W     3 W  U¦ W    3   [5] (a) You want to make as much money as possible, of course. Which stock would you buy? [10] (b) Suppose you decide to buy “Ystock” and your friend decides to buy “Zstock”. What is the probability that you make more money? (In other words, find ). ¨¡ b 7 ¤ a 5. Convergence of random sequences: ¤ ¥ 1WB YXHF cW ¤   1WB YXHF ¥¨  1 ¨ HF B ¢ ¨ ¢ £B ¡¤ B [10] (a) Let the probability space be given by , (restricted to , of course), and . For , let . Does this sequence of random variables converge? If so, to what and in what sense (almost surely, in probability, in mean square, in distribution)? If not, just establish that it does not converge to a limit in one sense (almost surely, in probability, in mean square, in distribution) - your choice! ) ¨ ¤  1 ¨ HF B ¢  ) ¨ ¤    a ¨ c ¦   ¨ ¨¦  g¨ ¨ ©B §¤ ¤ a ¢ £B ¡¤ B [10] (b) Let the probability space be given by , (restricted to , of course), and . For , let . Does this sequence of random variables converge? If so, to what and in what sense (almost surely, in probability, in mean square, in distribution)? If not, just establish that it does not converge to a limit in one sense (almost surely, in probability, in mean square, in distribution) - your choice! )   (restricted to ) ¤ ¥ ¢ , 1WB YXHF 1WB YXHF   ) ¤   , of  I % ¨ H¦ ¨   ¨ a     ( I  ¨ c ¦ ¨ ¨¦ ¤ A ¨ ¨ ©B §¤ ¤ a "" "" "" "" "" "" "" "" "" "" # "" ¨  ¤  "" ( B  a "" B 3 ( % ¨¦ ¤ ` ¨ ¨ ©B §¤ ¤ "" && ''&  ¨ " !"" B  3 % c ¢ £B ¡¤ B B HF   a F %  I  I 3 be given by , let ) [10] (c) Let the probability space course), and . For ""  "" "" "" "" "" && ''& W %  %     B    "" "$ Does this sequence of random variables converge? If so, to what and in what sense (almost surely, in probability, in mean square, in distribution)? If not, just establish that it does not converge to a limit in one sense (almost surely, in probability, in mean square, in distribution) - your choice! 6. Your team wins each match with probability 0.50 and loses each match with probability 0.50 (there are no ties). Assume that each match is independent of all other matches. Answer the following questions using the Central Limit Theorem. [10] (a) Assuming that the season lasts 100 matches, find the probability that your team wins more than 55 matches. [10] (b) Suppose that you wanted it such that your team wins at least 65 matches with probability 0.90. What is the minimum number of matches in a season for this to be true? Hint: Feel free to use the Central Limit Theorem loosely as on the homework to solve this part. ...
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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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