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Unformatted text preview: actual probability P ( | X- X | k X ). This shows that Chebyshevs 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 and variance 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 = 0. Find the distribution of Y = 1 /X . 8. Suppose that X has a Gamma distribution with parameters and . Show that Pr ( X 2 ) (2 /e ) . 9. Let X 1 , X 2 , . . ., X k be independent normal distributions with zero mean and unit vari-ance. Show that Y = k i =1 X 2 i has a Chi-squared distribution with degrees of freedom equal to k ....
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This note was uploaded on 12/05/2010 for the course GSB 603 at Stanford.