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Unformatted text preview: A sequence of random variables Z 1 ,Z 2 ,Z 3 ,... is said to converge in probability to some constant c if, for every strictly positive number ε > 0, we have lim n →∞ P (  Z n c  > ε ) = 0 . 9 In words, this means that for any positive ε we can think of – no matter how small – the probability that Z n differs from c by more than ε will eventually decrease to zero as n increases toward infinity. If Z 1 ,Z 2 ,Z 3 ,... converges in probability to c , we say that Z n → p c as n → ∞ . Returning to our example where Z n ∼ U (1 1 /n, 1+1 /n ), let us verify that in fact Z n → p 1. Choose any number ε > 0. We need to show that lim n →∞ P (  Z n 1  > ε ) = 0. Since Z n ∼ U (1 1 /n, 1+1 /n ), it must lie in the interval [1 1 /n, 1+ 1 /n ] with probability one. If n > 1 /ε , then the interval [1 1 /n, 1 + 1 /n ] lies strictly inside the interval [1 ε, 1+ ε ]. Therefore, we will have P (  Z n 1  > ε ) = 0 for all n > 1 /ε . No matter how small we chose ε , as n → ∞ we will eventually have n > 1 /ε . This shows that lim n →∞ P (  Z n 1  > ε ) = 0. Let us consider another example that is slightly less trivial. Suppose this time that the n th random variable in our sequence, Z n , is continuous with pdf f n given by f n ( x ) = (2 n 1) x 2 n for x ≥ 1 for x < 1 . The pdf f n is highest when x = 1, where it is equal to 2 n 1. As we move rightward along the xaxis from one, f n ( x ) decays smoothly to zero at the rate x 2 n . We can see that when n increases, f n spikes higher and sharper at one, and decays more rapidly to zero as we move along the xaxis. In a sense, the sequence of pdfs f n are gathering into an infinitely tall spike at x = 1 as n → ∞ . This is what we would expect if Z n → p 1, which is in fact the case, as we will now show. Choose any number ε > 0. Since f n ( x ) = 0 for x < 1, it must be true that P (  Z n 1  > ε ) = P ( Z n > 1 + ε ). Therefore, P (  Z n 1  > ε ) = Z ∞ 1+ ε (2 n 1) x 2 n d x = x 1 2 n ∞ 1+ ε = (1 + ε ) 1 2 n . Since ε is strictly positive, lim n →∞ (1 + ε ) 1 2 n = 0. It follows that Z n → p 1. 9 The law of large numbers Suppose we have an infinite sequence of random variables X 1 ,X 2 ,X 3 ,... . Assume that this sequence of random variables is independently and identically distributed , or iid. This means that (1) the different X i ’s are independent of one another, and (2) the different X i ’s all have the same distribution – that is, the same pmf or pdf. A consequence of (2) is that the different X i ’s all have the same expected value and variance. We will denote the common expected value by μ and the common variance by σ 2 . 10 For any n , we can imagine taking the average of the first n of the X i ’s. This average will be denoted ¯ X n : ¯ X n = 1 n n X i =1 X i ....
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 Spring '08
 Stohs
 Normal Distribution, Probability theory, probability density function

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