ece603_midterm2_2008

ece603_midterm2_2008 - ECE 603 - Probability and Random...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ECE 603 - Probability and Random Processes, Fall 2008 Midterm Exam #2 November 13th, 6:00-8:00pm, Agricultural Engineering 119 Overview The exam consists of five 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. and have joint probability density function ¡ 1. The random variables  A( " #!©§ ¦¤¢    ¨ ¥ £ 4 0 8 8 0 4 0  0 ( $ $ BA@ 9765321)¤'&% otherwise [8] (a) Find the value of . " . Y¨ 8 C X  VU ( WAT ¡  4U cbA( e gf¨ as: (  D  A( $ 6' else d  r¨ q s¦a¢ , the probability density function of . p !ih¨  d . Find 8X a%`E 4 004 BS2RQE  ¡d ¨  p Let such that D 8 G  I¨ § H£ ¢ [10] (e) Define the function . Given this information, find the  C [7] (d) Somebody tells you that is maximized. given . For your in terms of . ¡ ¨ D )¡ [10] (c) Find , the conditional probability density function of limits (which you should not forget), use and then bound   P  F4 E [5] (b) Write an expression (no need to evaluate) for 2. Jointly Gaussian random variables: , u vt ( # w ¡ vt u  ’‰ u vt ‘ 9‡ ¨ C ¡ ‰ E T ˆ ¡ ‚ † „¥ …ƒ§ ¦£ ‡ y  w $ ¡ €t u [10] (b) Let and be jointly Gaussian random variables. Suppose you know that , and that and are independent. By doing measurements, you find that and that . Find possible values for the pair ( , ), the variances of respectively, and (do not forget this part) the correlation coefficient . § ¦£ ¥ $§ $£ – ¡ – V • w $ ¡ ( B w ” ©$ ¡ ¡ €t u u vt ‚  •— , and , and , (where the Borel field is D~ D 45l 0 „ i D ~ 40 5l i }0 „ D ~ „0 !™l i0 }|(   d — €™˜¨ C $ $  u zw u t ” „ y $ u zw u t y$  {zxvt uy wu w eA( 4 oq r    l  i ¨ pnmkj¨ w ‚ u C qs u €t . Find u $ Ti¨ € € Let g hfd w eA( 4 3. [10] Let the probability space be given by restricted to , of course) and, for any interval, , ¡ ¡ u vt yT w $ $ u vt w ( “ ` w ( x ` . , be jointly Gaussian random variables. Let . Define . Find y # w $ [10] (a) Let and , and . 4. Counterexamples:  ¨ˆ‡… ¤’&† w 6mA( „ˆ [10] (a) Let the random variable be uniformly distributed on . Define and . Show that and are uncorrelated but not independent. Be rigorous here, particularly for the “not independent” part. ƒ u ƒ ¡ converges in mean square to but does not converge with , and the definition of the random variables .) €  ¨ ‚  C v™˜¨ d — Œ Ž k|  ƒ ¨ Š‰ ˆ @‹Y ¡ [10] (b) Give an example where probability one. (Be sure to give 5. Convergence: w eA( 4 d w eA( 4 g … d w eA( 4 g  ¨ | €   )—  €  d — €™˜¨ [10] (a) Let the probability space be given by , (restricted to , of course), and . For , let . Does this sequence of random variables converge in some sense? If so, to what and in what senses (almost surely, in probability, in mean square, in distribution)? As always, be sure to justify your claims. If you claim it does not converge in any sense, 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 u  , w eA( 4 £Q4 ¢ uE  P— u , of „ „ $   „  „ € D ¤ ¥€ $ C — be given by , let u ” 4 C ¡ i El }™h &mkj¨  l i¨ — €  d — €™˜¨ C [10] (b) Let the probability space course), and . For ¡ i El   l i¨ nmzj¨ C !I¨  € ”ˆ Does this sequence of random variables converge in some sense? If so, to what and in what senses (almost surely, in probability, in mean square, in distribution)? As always, be sure to justify your claims. If you claim it does not converge in any sense, just establish that it does not converge to a limit in one sense (almost surely, in probability, in mean square, in distribution) - your choice! (where the Borel field is  d w eA( 4 , g # u  B— D$ D 45l 0 „ i $ D 45l 0 i }0 „ $ D „0 !™l i0 }|(  c4 E ¨ € € ¨    d — €™˜¨ [10] (c) Let the probability space be given by restricted to , of course) and, for any interval, C $ $  uxyzwxvt ” „ u$ uy wu {z$ xvt  u zw u t y w eA( 4 oq r qs   l i¨ nmkj¨ u C A sequence of random variables is defined as follows. If is a rational number, for all . If is an irrational number, for all . Does this sequence of random variables converge in some sense? If so, to what and in what senses (almost surely, in probability, in mean square, in distribution)? As always, be sure to justify your claims. If you claim it does not converge in any sense, just establish that it does not converge to a limit in one sense (almost surely, in probability, in mean square, in distribution) - your choice! ¦ u ¢ € R¨  € ¦ ...
View Full Document

Ask a homework question - tutors are online