hw3_fall10 - ECE 603 Probability and Random Processes Fall...

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Unformatted text preview: ECE 603 - Probability and Random Processes, Fall 2010 Homework #3 Due: 10/8/10, in class 1. I flip a fair coin twice (assume the flips are independent) and record the outcome of these flips in order. For example, for “head followed by head”, the outcome is “HH”. Your job is to define a probability model that will used by one of your co-workers to analyze the experiment. You are not aware of what questions he/she might want to ask, so you want to generate as complete a model as possible (e.g. you do not want to use a trivial ). Provide a probability space for this experiment. Since the size of the sets involved here is not that large, be explicit about how each of these three things are defined. In particular, write out all of the sets in and give the probability of each. ¨¦ ©§¤ ¤¢ ¥£¡ 2. 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:   , the complement of ¤   , necessarily count-  ('&% $!$ ; that is, if and only if . Find the probability of . must contain an and ) 0! 4 31 2  ##! "  ¤   be the set of irrational numbers in , . Is  ¤   GE E E ¤ D¤ A¤  ¤  7 " [email protected]98PI  (c) Let for all be an arbitrary uncountable subset of  (b) Let able? -algebra. Show that  (a) Let be a subset of that is not in the Borel uncountable number of elements. GE E E ¤ D¤ A¤  ¤  7 " [email protected]9865 3. I have analyzed two independent experiments, Experiment 1 and Experiment 2, to arrive at two separate probability spaces: and , where: 1 ¨GA7 caXr¡ V ¦ , AE v 1   1 ¨ b¡ S q£Xp¦ ¨b q£¡ V ¦ is defined by , ¨ G  7¡ S cc`rXU¦ ¨g ¡ S ihXU¦ ¨g ih¡ V ¦ ¤S Q FTR¡  ¨Q wR¡ S ¦ 1 GGA¤7¤GA7¤G7¤ b `[email protected]`efaXefc`edc7 S 1  ¨Q wR¡ V ¦ 1 V G G s¤ A7¤ G s7¤ G A7¤ b `xCXefefaXedc7 1 , and . is defined by 1 ¨S ¦¤ U§S , and . sE t ¤V Q XWR¡ , uvE ¨GA7 caXr¡ S ¦ 1 GA¤ [email protected]`7 S YQ 1 , , ¨V ¦¤ 9¥§@V , , , yvE ¨Gs7 cr¡ V ¦ 1 VQ G s¤ A xCX7 1 (a) Are these valid probability spaces? Be sure to tell me all of the conditions that you checked to arrive at your answer. (b) My boss asks me to define a combined experiment as follows: Perform Experiment 1 and remember the result; then, perform Experiment 2 and remember the result. Now, write in your notebook the outcome of the combined experiment as an ordered pair with the first entry equal to the result of Experiment 1 and and the second entry equal to the result of Experiment 2. For example, an outcome might be “(1,4)”. Find for the combined experiment. Use a that captures as many events as possible, and be sure to write out explicitly at least half of the events in . ¨¦ €§¤ ¤Q YR¡ (c) Alas, the boss is fickle and changes his mind. Now he asks: perform Experiment 1 and remember the result; then, perform Experiment 2 and remember the result. Now, write in your notebook the equal to the result of Experiment 1 outcome of the combined experiment as a random variable times the result of Experiment 2. For example, an outcome might be “4” (which is ). Find  A‚ 0#A . Hint: Feel free to define ¦ for the random variable this makes it easier to represent. using the integral of a function if  ¨¦ €§¤ ¤ Q R¡ Now, your buddy in the modeling department comes to you with yet another experiment description: , where ¨ ¡ §¤ ¡ ¦ ¦¤ § 1 ¨b q£¡ ¦ is defined by ¨G7 cc`r¡ , and ¨g ih¡ ¡ ¦  ¨GD¤ A¤7 [email protected]`r¡ 1 G CB[email protected]`efaBCXefc`e¤ c7 G D¤ A¤7¤GD¤ A7¤G7 b . 1 1 ¡ ¦ §¤ ¢ , E  1 GD¤ A¤ [email protected]`7 1 ¨GD¤ A7 caBCXr¡ ¤¡ Q R¡ ¡Q ¦¤ E §@  (d) Is a valid probability space? Be sure to tell me all of the conditions that you checked to arrive at your answer. ¦ ¡ §¤ ¡ ¨ , of course, and ¦ ¨A¤  fBH¡ is restricted to ¨g ih¡ is V ¡ $§ Y ¥ $ ¡ V ¥$ % $ A §  $ A §  £ ¡ V  $¥ ¤¨ ¤¨ V ¤¨ ¥ ¨¦ €§¤  V V V  V ¥ V ¥ § ¡ £ ¨ A¤ ¡ ¤¤ fBHx¡ § ¡ © " #!  § ¡ ¡   1 ¨¨ § ¥¡ xF¨¤ ¦x¡  be the outcome of the experiment.  ¦ ¨ 9(%  ¨ fA  % S $   V¡ V ¡¡ ¦ ¦ (b) What is Q R¡ (a) What is ¤¡ where is a constant. Let , where 4. Consider the probability space defined by: to find the probability that a “3” is ¨ ¡ §¤ ¡ ¦ Q ¡ R¡ ¤ (e) Your boss asks you to use the description of observed. How do you respond? ? Your answer should be a number. ? Be sure to derive everything from first principles! (c) Since , it can be treated as a random variable. Find and roughly sketch the cumulative and probability density function of . distribution function (CDF) & '"  ¨ q! ¡ ) 21 ¨ q! ¡  4V  ) 0( (d) Let the random variable be defined as . Find the probability space for . Hint: This looks like a “function of random variable” problem, which we have not studied yet, but you can still easily solve it from first principles. ¨ 5¦ 6§¤ ¤ £¡ ¢ A  1 3 3 3 ...
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