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Unformatted text preview: CS 70 Discrete Mathematics for CS Spring 2005 Clancy/Wagner Notes 21 Some Important Distributions Question : A biased coin with Heads probability p is tossed repeatedly until the first Head appears. What is the expected number of tosses? As always, our first step in answering the question must be to define the sample space . A moments thought tells us that = { H , TH , TTH , TTTH , . . . } , i.e., consists of all sequences over the alphabet { H , T } that end with H and contain no other H s. This is our first example of an infinite sample space (though it is still discrete). What is the probability of a sample point, say = TTH ? Since successive coin tosses are independent (this is implicit in the statement of the problem), we have Pr [ TTH ] = ( 1 p ) ( 1 p ) p = ( 1 p ) 2 p . And generally, for any sequence of length i , we have Pr [ ] = ( 1 p ) i 1 p . To be sure everything is consistent, we should check that the probabilities of all the sample points add up to 1. Since there is exactly one sequence of each length i 1 in , we have Pr [ ] = i = 1 ( 1 p ) i 1 p = p i = ( 1 p ) i = p 1 1 ( 1 p ) = 1 , as expected. [In the secondlast step here, we used the formula for summing a geometric series.] Now let the random variable X denote the number of tosses in our sequence (i.e., X ( ) is the length of ). Our goal is to compute E ( X ) . Despite the fact that X counts something, theres no obvious way to write it as a sum of simple r.v.s as we did in many examples in the last lecture. (Try it!) Instead, lets just dive in and try a direct computation. Note that the distribution of X is quite simple: Pr [ X = i ] = ( 1 p ) i 1 p for i = 1 , 2 , 3 , . . . So from the definition of expectation we have E ( X ) = ( 1 p ) + ( 2 ( 1 p ) p ) + ( 3 ( 1 p ) 2 p ) + = p i = 1 i ( 1 p ) i 1 . This series is a blend of an arithmetic series (the i part) and a geometric series (the ( 1 p ) i 1 part). There are several ways to sum it. Here is one way, using an auxiliary trick (given in the following Theorem) that is often very useful. [Ask your TA about other ways.] Theorem 21.1 : Let X be a random variable that takes on only nonnegative integer values. Then E ( X ) = i = 1 Pr [ X i ] . CS 70, Spring 2005, Notes 21 1 Proof : For notational convenience, lets write p i = Pr [ X = i ] , for i = , 1 , 2 , . . . . From the definition of expectation, we have E ( X ) = ( p ) + ( 1 p 1 ) + ( 2 p 2 ) + ( 3 p 3 ) + ( 4 p 4 ) + = p 1 + ( p 2 + p 2 ) + ( p 3 + p 3 + p 3 ) + ( p 4 + p 4 + p 4 + p 4 ) + = ( p 1 + p 2 + p 3 + p 4 + ) + ( p 2 + p 3 + p 4 + ) + ( p 3 + p 4 + ) + ( p 4 + ) + = Pr [ X 1 ] + Pr [ X 2 ] + Pr [ X 3 ] + Pr [ X 4 ] + ....
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This note was uploaded on 09/03/2011 for the course CS 70 taught by Professor Papadimitrou during the Fall '08 term at University of California, Berkeley.
 Fall '08
 PAPADIMITROU
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