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Problem Set 2 in Econ. 240A, Spring 1998 by McFadden, Exercises of Econometrics and Mathematical Economics

Problem set 2 from economics 240a, spring 1998, taught by mcfadden. The problems cover various topics in probability theory and statistics, including geometric distributions, characteristic functions, log normal distributions, and extreme value distributions. Students are asked to find expected values, probabilities, and moment generating functions.

Typology: Exercises

2010/2011

Uploaded on 10/31/2011

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Download Problem Set 2 in Econ. 240A, Spring 1998 by McFadden and more Exercises Econometrics and Mathematical Economics in PDF only on Docsity! Econ. 240A, Spring 1998 D. McFadden PROBLEM SET 2 (Properties of Special Distributions) (Due Monday, Feb. 16, with discussion in section on Feb. 11) 1. Suppose that the duration of a spell of unemployment (in days) can be described by ka geometric distribution, Prob(k) = p (1-p), where 0 < p < 1 is a parameter and k is a non-negative integer. What is the expected duration of unemployment? What is the probability of a spell of unemployment lasting longer than K days? What is the conditional expectation of the duration of unemployment, given the event that k > m, where m is a positive integer? 3 42. Using the characteristic function, find EX and EX for a standard normal X. 3. A log normal random variable Y is one that has log(Y) normal. If log(Y) has mean 2µ and variance σ , find the mean and variance of Y. [Hint: It is useful to find the moment generating function of Z = log(Y).] 4. If X and Y are independent normal, then X+Y is again normal, so that one can say that the normal family is closed under addition. (Addition of random variables is also called convolution, from the formula for the density of the sum.) Now suppose X and Y are independent and have extreme value distributions, a-x b-yProb(X ≤ x) = exp(-e ) and Prob(Y ≤ y) = exp(-e ) , where a and b are location parameters. Show that max(X,Y) once again has an extreme a bvalue distribution (with location parameter c = log(e +e )), so that the extreme value family is closed under maximization. 25. If X is standard normal, derive the density and characteristic function of Y = X , and confirm that this is the same as the tabled density of a chi-square random variable with one degree of freedom. If X is normal with variance one and a mean µ that is not zero, derive the density of Y, which is non-central chi-square 2distributed with one degree of freedom and noncentrality parameter µ . 1
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