ece603_midterm1_2002

ece603_midterm1_2002 - ECE 603 - Probability and Random...

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Unformatted text preview: ECE 603 - Probability and Random Processes, Fall 2002 Midterm Exam #1 October 17th, 6:00-8:00pm, Marston 132 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 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. 1. A number is chosen at random from the interval [0,1]. As is the standard case, the probabilities are defined on the Borel -algebra (restricted to [0,1]). Starting from first principles (i.e. definition of the Borel -algebra, axioms of probability, etc.), answer the following three parts: ¡ [10] (a) Let be a subset of that is not in the Borel -algebra. Show that must contain an uncountable number of elements. ©§¥£ ¨¦¤¢ . Is , the comple ©§¥£ ¨¢ ¡ [5] (b) Let be an arbitrary uncountable subset of ment of , necessarily countable?   ,  @9 ; that is,    ©§¥£ ¤¢ [10] (c) Let be the set of irrational numbers in if and only if and for all . Find the probability of .  76 6 6 ¥ 5 ¥ 3 ¥ § ¥ £ 0 8¨¨4¦42¨1 (' ) $% & § #"!   £  76 6 6 ¥ 5 ¥ 3 ¥ § ¥ £ 8¨¨4BA2¨10 2. [10] A number is chosen from the interval [0,1] such that the likelihood of a given result is proportional to the value . Define a non-trivial probability , where is the observation space for this experiment; that is, find space, is a set of subsets of to which probabilities are assigned, and is a probability mapping from to .  D  Q R4IHFEC P¥ G¥ D P D ©§¥£ ¨¢ G G 3. Let the continuous random variable be uniformly distributed between and . Let the discrete random variable have probability mass function defined by , , and . Form the random variable . Assume and are independent. £ S T & TC cbP T T gS Q V§ & TC EUP f h if Q tC us1q r © w¤Uv f¢ T f TC EUP S & and Var & Q 23 Y W X& , the probability density function of 3 § & , what is the probability that . Q d5 a W `& [10] (b) Find © R¤¢ f W p [10] (c) Find e W X& [10] (a) Given that . . have joint probability density function: § h S h "f h 6 £ and f § ¨ ¥Q AV§ S 4. The random variables else ¥ ¦¤ C & Qt H¥ C r ¥ £ ¢ £¡q ¥ [5] (a) Compute the constant . t  uC r q © &  S f ¤¢ and Var  © &  S f ¢ v [10] (d) Compute at . . Q ©q F C [5] (c) Compute the conditional probability density function Q [5] (b) Compute the marginal probability density function . . Interpret your answers £ ¢ Uv , the cumulative distribution function "! #£ . Compute . Q dC W &   W f S &   [10] (f) Let (CDF) of . . Find © & [5] (e) Let T T 666¥3 ¨¨¦A¥ be mutually indepen- 6 £ 8 9S &$ '¥ %S 6 7 £ h 0 420 531¤ ¥ &  , ..., . Find Q tC us1q r & § Q C Y ¥ ¦£ S , ( )¥ W S ¡q and define as the maximum of bility density function of . 7 5. [15] Let the random variables dent and each have probability density function: , the proba- f f ...
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