n13 - CS 70-2 Discrete Mathematics and Probability Theory...

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Unformatted text preview: CS 70-2 Discrete Mathematics and Probability Theory Spring 2009 Alistair Sinclair, David Tse Note 13 Random Variables and Expectation Question : The homeworks of 20 students are collected in, randomly shuffled and returned to the students. How many students receive their own homework? To answer this question, we first need to specify the probability space: plainly, it should consist of all 20! permutations of the homeworks, each with probability 1 20! . [Note that this is the same as the probability space for card shuffling, except that the number of items being shuffled is now 20 rather than 52.] It helps to have a picture of a permutation. Think of 20 books lined up on a shelf, labeled from left to right with 1 , 2 , . . . , 20. A permutation is just a reordering of the books, which we can describe just by listing their labels from left to right. Lets denote by i the label of the book that is in position i . We are interested in the number of books that are still in their original position, i.e., in the number of i s such that i = i . These are often known as fixed points of the permutation. Of course, our question does not have a simple numerical answer (such as 6), because the number depends on the particular permutation we choose (i.e., on the sample point). Lets call the number of fixed points X . To make life simpler, lets also shrink the class size down to 3 for a while. The following table gives a complete listing of the sample space (of size 3! = 6), together with the corresponding value of X for each sample point. [We use our bookshelf convention for writing a permutation: thus, for example, the permutation 312 means that book 3 is on the left, book 1 in the center, and book 2 on the right. You should check you agree with this table.] permutation value of X 123 3 132 1 213 1 231 312 321 1 Thus we see that X takes on values 0, 1 or 3, depending on the sample point. A quantity like this, which takes on some numerical value at each sample point, is called a random variable (or r.v. ) on the sample space. Definition 13.1 (random variable) : A random variable X on a sample space is a function that assigns to each sample point a real number X ( ) . Until further notice, well restrict out attention to random variables that are discrete , i.e., they take values in a range that is finite or countably infinite (such as the integers or the rationals), rather than the real numbers. The r.v. X in our permutation example above is completely specified by its values at all sample points, as given in the above table. (Thus, for example, X ( 123 ) = 3 etc.) Rather than the value at each sample point, we are usually more interested in the set of points at which the CS 70-2, Spring 2009, Note 13 1 r.v. takes on some given value. Let a be any number in the range of X . Then the set { : X ( ) = a } is an event in the sample space (why?). We usually abbreviate this event to simply X = a . Since....
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n13 - CS 70-2 Discrete Mathematics and Probability Theory...

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