midterm1_fall2010

midterm1_fall2010 - ECE 603 - Probability and Random...

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Unformatted text preview: ECE 603 - Probability and Random Processes, Fall 2010 Midterm Exam #1 October 20th, 6:30-8:30pm Overview The exam consists of five 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 one 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. Hint: You may find the following fact useful as you solve this exam: $ 7$CB!A @987 5432§1)'&%#"  ©§¥  6 0 ¥ ( $  !    ¨ ¦ ¢ ¢¤ £¡ 1. Consider the network: F F TF G F S U F F QF G F P E E F F ¨ FG F E E F F IF G F R D F F IF G F H WG and let denote the probability that switch is closed (i.e. that packets can flow through that switch). Now, suppose that the events have the following properties: RG S fG . . RV V €¨ V  TH ys V ts u¨ V TH ( V  g$ ¨ q ($ H v7$ P xbqf¨ wv7$ P i ($ rdfph$ P qi ( form a partition. and .  $  $ H V . in terms of unions and intersections of the and $P V $ 7$ P V and can be re-designed, but $  $  V $P V $ 7$ P V  t ¨ V  TH V  s t ¨ V TH V s $ 7$ P V t ¨ V  s H V  How would you specify  y ¨ V  TH V  s y ¨ V TH V s $ 7$ P V y ¨ V  s H  How would you specify and is closed. such that such that $ H V [10] (d) Suppose that remain the same as in (a). H ƒG  occurs, find the probability that ’s (and their comple- W ‚V SV V V V [8] (b) Write an expression for ments), and find . [5] (c) Given that D ¨V be the event that there is a connection between [7] (a) Find and . U PV and P QG  ¨ G are independent. are independent of the status of switches HG  d b a Y ( X W `c7Q`Q` C7 CV Let and and W eV The status of switches must is maximized? is minimized? , of  ¢ ( ¥ ¦ ¤ £  fi (  D $   ( 7$  G 7 $ $    PG P ¢ ©  ¡   by: (restricted to £ ,  fi [10] (a) Define the random variable is defined by ¤ 2. Suppose that the probability space . course), and § § ¨    i Y !  U $ PH $  D  ? © % & ' Apple, Banana, Cardinal, Dove b  4 5  Y        Y b  !4 65  ! i !   !  !    Qa Y #$"   ( g$    D D   U   ( g$  1 )11 0 11   13 & Apple, Cardinal . Find ( ( 8 & Apple, Banana and 8 ( ( 7 Define the events 7 U 2 Find the induced probability space for (feel free to use the first letters “A”, “B”, “C”, and “D” of Apple, Banana, Cardinal, and Dove, respectively, to save you space) $s [10] (b) Consider the mapping (not a random variable!) given by (     . falls in the interval Apple Banana Cardinal Dove   What is the probability for Y  Find the probability density function . 3. Suppose that I conduct the following experiment. I flip a coin continually, and I write down the result of the flips to get an outcome . For example, a possible might look like:  q$ q q of all possible outcomes . Is 9   $ ¢ 9 (  [10] (a) Consider the sample space Tail, Tail, Tail, Head, Head, countable or uncountable? [10] (b) Construct a rich (i.e. non-trivial) probability space for this experiment. You should end up with an with an uncountable number of elements that should allow one to answer any question of interest. 9 ¢ D be a random variable with probability density function:   i    ( g$   f i #" be a random variable with probability density function ¥ ¥  $ Y  ¤  Ya ! $  ¤  a g$ ¤  ( ¡ and let else  4. Let ¢ £" Answer the following six parts independently. In each case, the answer and a single line of justification is sufficient. .  Y  fi  Qd  ( v$  ¨ $ § . . ¦ £" . Find $ § 4 P ! ¡£ ¡  ¤ and ¤ ! P ¡ ¡£ b )(  ¦ " [7] (d) Find U [8] (e) Let   Find else  ¦ £" , where . Y $ § $  ©( ¨ D U ( D U Y! ( DU [5] (c) Let . Find ¦ £" [5] (b) Let . Find $ § [5] (a) Let . [5] (f) Suppose that I run 10 trials, each resulting in an independent observation of a random variable with probability density function . What is the probability that the sum of the outcomes exceeds 15? $ ¤ ¢ " D has probability density function:        ¨¦¥¤¢ ¡h$  ©§£ ( #" which is shown below (Be sure to look at the plot!).  5. The random variable [10] (a) Find . D If you cannot get part (a), proceed now do do parts (b) through (d) as if is Gaussian with mean and variance . Make sure to note on your paper that you are doing such.  PD ¤ a £ . (Hint: You do not have to do any integrals - look at your equation for finding the [5] (b) Find variance of a random variable.) . ¤   D Y! b! d PD D . 0.25 0.2 f(x) 0.15 0.1 0.05 0 −10 −8 −6 −4 −2 0 x 2 4 Figure 1: Probability density function of 6 D £  [5] (d) Find $  [5] (c) Find 8 10 for Problem 5. ...
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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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