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Week_9_Slides - Week 9 F Distribution Inequalities...

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Week 9 F – Distribution Inequalities Convergence
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F distribution and F statistic 2
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Chebyshev’s Inequality 3 Let k > 0 and let Y be a rv with mean and variance 2   . Then 2 1 (| | ) P Y k k and 2 1 (| | ) 1 P Y k k Proof : 2 2 2 2 ( ) {( ) (| | )} {( ) (| | )} E Y E Y I Y k E Y I Y k 2 {0 (| | )} {( ) (| | )} E I Y k E k I Y k 2 2 (| | ) k P Y k . Applications: Used to prove many probabilistic results including the weak law of large numbers.
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Markov’s Inequality 4 For any rv U and real numbers x , p > 0, (| | ) | | p p P U x x E U .
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Bivariate Normal 5 The multivariate normal distribution Suppose that the pdf of 1 ( ,..., ) k Y Y Y is given by 1 /2 1/2 1 1 ( ) exp ( ) ( ) (2 ) | | 2 n f y y y , 1 ( ,..., ) k k y y y   , where 1 ( ,..., ) k k   and 11 12 1 21 22 2 1 2 k k k k
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