P Set 2 - actual probability P ( | X- X | k X ). This shows...

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1 MGTECON 603, Econometrics I Jules H. van Binsbergen Fall 2009 Stanford GSB PROBLEM SET 2 Due: Friday October 9 th In Class 1. Suppose that the random variable X has an exponential distribution with pdf f X ( x ) = exp( - x ), x > 0, and 0 elsewhere. (a) Find the pdf for Y = 1 /X . (b) Find the pdf for Y = ln( X ). (c) Find the pdf for Y = 1 - F X ( X ). 2. Let f ( x ) = 1 / 3 for - 1 < x < 2 and zero elswhere be the pdf for a random variable X . Find the pdf and distribution function for the random variable Y = X 2 . 3. Suppose that the random variable X has cdf F X ( x ) = 0 if x < 0 1 / 2 if x = 0 , ( x + 1) / 2 if 0 < x 1 , 1 if x > 1 . Is X a discrete random variable? Is X a continuous random variable? Calculate the mean and variance of X . 4. Let X be a random variable with moment generating function M X ( t ), - h < t < h . Prove that P ( X a ) exp( - at ) · M X ( t ) , 0 < t < h, and P ( X a ) exp( - at ) · M X ( t ) , - h < t < 0 . 5. Let X be a discrete random variable with P ( X = - 1) = 1 / 8, P ( X = 0) = 6 / 8 and P ( X = 1) = 1 / 8, and P ( X = c ) = 0 for all other values of c . Calculate the bound on P ( | X - μ X | ≥ k · σ X ) for k = 2 using Chebyshev’s inequality. Compare this to the
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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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P Set 2 - actual probability P ( | X- X | k X ). This shows...

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