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Random Variables - Probability - Solved Exam

Exams, Probability

Post: February 27th, 2013
Description
This is the Solved Exam of Probability which includes Exponentially Distributed, Continuous, Random Variable, Expected, Positive Integer, Geometric Distribution, Probability, Geometrically, Distributed, Value etc. Key important points are: Random Variables, First Line, Amount, Assignment, Cauchy Distributed, Independent, Density Function, Partial Fractions, Convolution Product, Logarithm Approach
This is the Solved Exam of Probability which includes Exponentially Distributed, Continuous, Random Variable, Expected, Positive Integer, Geometric Distribution, Probability, Geometrically, Distributed, Value etc. Key important points are: Random Variables, First Line, Amount, Assignment, Cauchy Distributed, Independent, Density Function, Partial Fractions, Convolution Product, Logarithm Approach
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Probability: Problem Set 6 Fall 2009 Instructor: W. D. Gillam Due Nov. 6, start of class Instructions. Print your name in the upper right corner of the paper and write “Problem Set 6” on the first line on the left. Skip a few lines. When you finish this, indicate on the second line on the left the amount of time you spent on this assignment and rate its difficulty on a scale of 1 − 5 (1 = easy, 5 = hard). (1) Let X, Y be Cauchy distributed independent random variables. Calculate the density function for X + Y . Solution. The density function for the Cauchy distribution is 1 f (x) = π (1 + x2 ) so the density for X + Y is given by the convolution product, which can be integrated by partial fractions: (f ∗ f )(x) = = = ∞ −∞ f (y )f (x − y )dy ∞ −∞ 1 π2 2 = π (4 + x2 ) = (1/2)f (x/2). 1 π2  1+y x(arctan y − arctan(x − y )) + ln 1+(x−y)2 1 1 dy 2 ) 1 + (x − y )2 (1 + y 4x + x3 2 ∞  −∞ Note that the terms in the logarithm appro..

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