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 continuous distribution that has cdf F ( x ) and pdf f ( x ). De±ne Z n = n [1F ( X ( n ) )]. Find the limiting distribution (convergence in distribution) of Z n ....
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 Fall '11
 Manuel
 Standard Deviation, Variance, Probability theory, unbiased estimator

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