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Unformatted text preview: actual probability P (  Xμ X  ≥ k · σ X ). This shows that Chebyshev’s inequality is indeed a sharp bound for some k for some random variables. Problem Set 2, MGTECON 603, 2009 2 6. Let X have a normal distribution with mean μ and variance σ 2 . Find the probability density function of Y = exp( X ). This is known as a lognormal distribution. 7. Suppose that X has a Cauchy distribution with θ = 0. Find the distribution of Y = 1 /X . 8. Suppose that X has a Gamma distribution with parameters α and β . Show that Pr ( X ≥ 2 αβ ) ≤ (2 /e ) α . 9. Let X 1 , X 2 , . . ., X k be independent normal distributions with zero mean and unit variance. Show that Y = ∑ k i =1 X 2 i has a Chisquared distribution with degrees of freedom equal to k ....
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 '09
 BIRSBERGEN
 Normal Distribution, Variance, Probability theory, Jules H. van Binsbergen Stanford

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