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Unformatted text preview: CS 702 Discrete Mathematics and Probability Theory Spring 2009 Alistair Sinclair, David Tse Note 14 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 Ω . A moment’s thought tells us that Ω = { H , TH , TTH , TTTH , . . . } , i.e., Ω 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 ω = 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 ω ∈ Ω of length i , we have Pr [ ω ] = ( 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 Ω , we have ∑ ω ∈ Ω Pr [ ω ] = ∞ ∑ 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 ( ω ) is the length of ω ). Our goal is to compute E ( X ) . Despite the fact that X counts something, there’s no obvious way to write it as a sum of simple r.v.’s as we did in many examples in an earlier lecture note. (Try it!) In a later lecture, we will give a slick way to do this calculation. For now, let’s just dive in and try a direct computation of E ( X ) . 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 14.1 : Let X be a random variable that takes on only nonnegative integer values. Then E ( X ) = ∞ ∑ i = 1 Pr [ X ≥ i ] . CS 702, Spring 2009, Note 14 1 Proof : For notational convenience, let’s 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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 Spring '08
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