This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: actual probability P (  XÎ¼ X  â‰¥ k Â· Ïƒ X ). This shows that Chebyshevâ€™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 Î¼ and variance Ïƒ 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 Î¸ = 0. Find the distribution of Y = 1 /X . 8. Suppose that X has a Gamma distribution with parameters Î± and Î² . Show that Pr ( X â‰¥ 2 Î±Î² ) â‰¤ (2 /e ) Î± . 9. Let X 1 , X 2 , . . ., X k be independent normal distributions with zero mean and unit variance. Show that Y = âˆ‘ k i =1 X 2 i has a Chisquared distribution with degrees of freedom equal to k ....
View
Full Document
 '09
 BIRSBERGEN
 Normal Distribution, Variance, Probability theory, Jules H. van Binsbergen Stanford

Click to edit the document details