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# hw4 - this to the bound you get from Chebyshev’s...

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APPM 4/5520 Problem Set Four (Due Wednesday, October 5th) 1. Let X 1 , X 2 , . . . , X n be a random sample from any distribution with mean μ and variance σ 2 . From class, we know that ˆ μ = X is an unbiased estimator of μ . Suppose that we want to estimate the variance. The interpretation of the variance ( σ 2 = E [( X - μ ) 2 ]) as the mean squared deviation from the mean μ leads us to the natural estimator that involves averaging the squared deviations in the sample from the sample mean: n i =1 ( X i - X ) 2 n . Some books/people refer to this quantity as the sample variance . However, many other books/people (including me!) define the sample variance to be n i =1 ( X i - X ) 2 n - 1 because it is an unbiased estimator for σ 2 . Let S 2 1 := n i =1 ( X i - X ) 2 n and S 2 2 = n i =1 ( X i - X ) 2 n - 1 . (a) Show that S 2 2 is an unbiased estimator of σ 2 . (b) Use part (a) to quickly find the expected value of S 2 1 . ( Hint: Don’t make this too much work! Use part (a)! ) 2. Let X exp ( rate = λ ). Find the exact value of P ( | X - μ X | ≥ X ) for any k >
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Unformatted text preview: this to the bound you get from Chebyshev’s inequality. 3. Consider the sequence of independent random variables X 1 , X 2 , . . . where X n ∼ bin ( n, p ). For i = 1 , 2 , . . . , de±ne Y n = X n /n . Investigate the convergence in probability of Y n . (i.e.: Say what it converges to and show it!) 4. Let X 1 , X 2 , . . . , X n be a random sample from the distribution with pdf f ( x ) = 1 x 2 I (1 , ∞ ) ( x ) . (a) Find the limiting distribution of X (1) = min( X 1 , X 2 , . . . , X n ). (b) Find the limiting distribution of n ln X (1) . 5. Let X ( n ) = max( X 1 , X 2 , . . . , X n ) where X 1 , X 2 , . . . , X n is a random sample from any contin-uous distribution that has cdf F ( x ) and pdf f ( x ). De±ne Z n = n [1-F ( X ( n ) )]. Find the limiting distribution (convergence in distribution) of Z n ....
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