831_09hw6

831_09hw6 - probabilities p, 1-p , in other words S n = X 1...

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831 Theory of Probability Fall 2009 Homework 6 Due Thursday, November 12 1. IID variables close up. Let { Y k } be i.i.d. random variables on R with a common continuous density f . As we put more and more points Y k down they tend to concentrate so let us spread them out by multiplying by n , and also look at them around the point nc , for some fixed c R . In other words, for each n N define X n,k = n ( Y k - c ) , 1 k n. Let N n ( a,b ) = n k =1 1 ( a,b ) ( X n,k ) be the number of X n,k that fall into ( a,b ). Find a weak limit for N n ( a,b ) as n → ∞ . 2. Exercises 1.3, 1.6 and 1.7 on p. 174–175. 3. Let 0 < p < 1 and let S n be the simple random walk with step
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Unformatted text preview: probabilities p, 1-p , in other words S n = X 1 + + X n and the { X i } are i.i.d. random variables with distribution P ( X i = 1) = p and P ( X i =-1) = 1-p. Fix a positive integer b &gt; 0, and let T be the rst hitting time of the point b : T ( ) = inf { n 1 : S n ( ) = b } . If the walk never hits b , in other words the set { n : S n ( ) = b } is empty, then T ( ) = . Calculate the expectation ET . Be sure to justify everything....
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