final_2010 - ECE 603 - Probability and Random Processes,...

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Unformatted text preview: ECE 603 - Probability and Random Processes, Fall 2010 Final Exam December 14th, 10:30am-12:30pm, Marston 132 Overview The exam consists of five problems for 110 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 three 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. ©“ A'„ T©“ U‘'„ ©„ 4E¥  ™ „ j ‘ u au ©W 1i¥ V h & h kg  pW j A u ©W ei¥ V u  „ E¥ ”   5 &  5s uu W ‚  5 ‘'„ ©“ 5 # ‘'„ ©“ h h ig sfHu u ‚ W sv fHu u W a ©„ $4E¥ ‘c ‰ d d™ e e"` `  # E¥ ”   „ # E¥ ” „ ©„ 4E¥ , ©„ 4E¥ is the Fourier Transform of sinc 7 ˜Y Y b—– ¦` ’ '€ ©W“ 1¢'„ „ ”R E¥ •S  q ©„ 4E¥ a  ©W $ei¥ V ¥£¡ y¤¢ €'ywu u xs ‚ W €xs v 'ywu u W aY b¦ Y` & ¥ 20 P31£ ’ '€ ©W“ 1¢'„ ƒ c  ‰d‡† …  © „  ‘c Dˆirp4E¥ d sinc cg  S) `C c fd Fe Parseval’s Relation: If s q  © W trp1i¥ h # X¨¥ V W7 © A@ Time Function Fourier Transform  A%9¨431£ © @ 8 7¥ 20 © @¥ 20 £ © 7¥ £ ¡ 8 © @¥ £ ¡ © 7¥ 20 AE431H¨¦¤¢GF¢ED¤¢C¨4B1£  © @ 8 7¥ £ ¡ A%9¨¦¤¢ © @¥ 20 £ © 7¥ 20 £ I © @¥ £ ¡ © 7¥ £ ¡ AE431H¨431GPAE'¤¢C¨D¤¢  QAE¦¤¢H¨¦¤¢ © @¥ £ ¡ © 7¥ £ ¡ 5 ¨¦¤¢ S  7¥ £ ¡ R # A@ ©  QAE431H¨431£ © @¥ 20 £ © 7¥ 20 5 ¨¦¤¢ R   7¥ £ ¡ 5 A@ ©  QAE¦¤¢H¨431£ © @¥ £ ¡ © 7¥ 20 5 ¨431£ S  7¥ 20 R # A@ © # ¨¦¤¢ 7¥ £ ¡ T© UA@ # ¨¦¤¢ 7¥ £ ¡ T© UA@ # ¨431£ 7¥ 20 T© UA@  ) ! (&    ¨¦¤¢ © §¥ £ ¡ )!& # !  ¢"('%$"   5 6!     ¨431£ © §¥ 20  Some potentially useful information have joint probability density function:  V v ˜  v v ¦ ˜ v 9s 5 ˜ ©q ¨ V © V ¦ ¥ £¡ §˜ S¥ ¤„ V ¦ ¥ £¡ §˜ S¥ ¤¢„  $© V ¦ §˜ ¥ © plane where 5 a [5] (a) Shade the region in the otherwise ¦ and 1. The random variables is non-zero. [5] (b) Find . ¨ , the marginal probability density function of ©V © V ¥ „ ¡ ¡Y ¥ „ ¦¥ , the conditional probability density function of given . For this, write between constants and your limits on as (potentially) functions of . a V ¦ u V [5] (d) Find your limits for . a [5] (c) Find 2. Consider the following experiment: I flip a fair coin. If the coin comes up “heads”, I record the outcome (zero) in my notebook; if the coin comes up “tails”, I generate a random number from a uniform distribution on the interval , and record the result as the outcome in my notebook. As has been the standard from class, denote the outcome of the experiment .  T ˜ s s bR 5   , . . , find the probability that the coin came up “heads”. '  # s   ©    ¥ ƒ $   %#"! © Given ƒ Find for the experiment. (If you have a hard time finding for partial credit.) ƒ  §˜ ¥ ˜ [10] (b) Let © [10] (a) Find a non-trivial space give the probability density function of  ¥ (& ' [10] (c) Let . 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. '& $ )¢! 3 ’s, each with mean  #s  1 2a 3. For a fixed , and independent and identically distributed sequence of and variance , we know that, if , then: A 91 7 1 5a  B@8 1  T 5a R 3 E0 4 0 1  T 5a R 4 9C  D1 A  9C 1  T 5a  B1 A R 4 T R 0 1 T 5a R 6 4 4 Answer each of the parts below independently. , but now . 15a  AB91 7 6  1 T Ga R 4 and variance . Again, let  s  R  3   [5] (a) Suppose that the are still identically distributed with they are correlated such that , for some in terms of , , , and . Find  (& 1 I R 6  6© 1 Fa ˜ R P H a 1Qa ¥ ¢I¨ 6 3 T 0 R 4 [10] (b) Now, go back to assuming an independent and identically distributed sequence of ’s, each with mean and variance . But now suppose that is also random. That is, suppose 1a 0  6 1 T Sa R 4  3 I have a non-negative discrete random variable ; I generate and then sum together the first elements of the sequence . Call the result . Find the expected value of in terms of , the variance of , and . (Hint: Write the joint probability density function of and as , and use the definition of the expectation of to get started. You can get to the answer in just a few steps if you head the right direction.) T 0 0 4 0R 0 0  6 $1 a! 3  0 © with mean zero and autocorrelation function ˜ ¥ sfu u ‚ sv fu u ¥ ¡ ©W 1i¥ ¡ ©W ei¥ ¥ ˜ 5s uu § ¨¡ q© 7  B1 A9 ©¥ 96 ¥ „ e© 9 ¦¥ u Y ¥„ 9 4. Consider a stationary Gaussian random process ¥£ ¦¥ ¤¢  [10] (a) Consider the sequence formed by averaging samples of taken one second apart. 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. ¡ formed by averaging samples of ©W 1i¥ © € 'x § ¨¥ ¡   B1 A9 & 7 9  9& 9& Find the probability density function of & $ 9 ¢! 9 [20] (b) Consider the sequence one-half second apart. taken . (Note: Starting from the probability density function of is probably not the easiest way to solve this part.) Determine whether the sequence converges, and, if so, to what and in what ways? Consider only 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.  ©W 1i¥ ¡ & & $ 9 ¢! 5. Suppose I have a zero-mean stationary Gaussian random process with autocorrelation function . Suppose is input to a linear time-invariant filter with impulse response . You measure the power at the output of the filter to be 3 milliWatts.  1i¥ ©W  ¨ €  ¨  ©  ©W 1i¥  ©W ei¥  © © ¡ ©W 1i¥ © ¥ ¦¥ ©ei¥ W 2  30 £ 30 £ 2 ¡ ©W 1i¥ ©W 1i¥ ¡ ©W ei¥  30 £ 2 [8] (b) Suppose is input to a linear time-invariant filter with impulse response . Find the power at the output of the filter.  ¤¨ [7] (a) Suppose is input to a linear time-invariant filter with impulse response . Find the power at the output of the filter. © s 5 i¥ 30 £ W2 ...
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