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Unformatted text preview: CS 70 Discrete Mathematics and Probability Theory Spring 2010 Alistair Sinclair Note 15 Some Important Distributions The first important distribution we learned about in the last Lecture Note is the binomial distribution Bin ( n , p ) . This is the distribution of the number of Heads in n tosses of a biased coin with probability p to be Heads. Its expectation is np . In this Note, we explore two other important distributions. The first one is also associated with a coin tossing experiment. Geometric Distribution Question : A biased coin with Heads probability p is tossed repeatedly until the first Head appears. What is the distribution and 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 , T H , T T H , T T T H , . . . } , 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 = T T H ? Since successive coin tosses are independent (this is implicit in the statement of the problem), we have Pr [ T T H ] = ( 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 ). Its distribution has a special name: it is called the geometric distribution with parameter p (where p is the probability that the coin comes up Heads on each toss). Definition 15.1 (geometric distribution) : A random variable X for which Pr [ X = i ] = ( 1 p ) i 1 p for i = 1 , 2 , 3 , . . . is said to have the geometric distribution with parameter p . This is abbreviated as X ∼ Geom ( p ) . If we plot the distribution of X (i.e., the values Pr [ X = i ] against i ) we get a curve that decreases monotoni cally by a factor of 1 p at each step. See Figure 1. CS 70, Spring 2010, Note 15 1 Figure 1: The Geometric distribution. Our next 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 , . . ....
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 Spring '08
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 The Land, Probability theory, Geometric distribution

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