midterm_solution_prob_thy

midterm_solution_prob_thy - MATHEMATICS 2603 Midterm Show...

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MATHEMATICS 2603 Midterm, October 21, 2009 Show all your work. Use back of page if necessary. You can use your own BLANK scratch paper. Name : ID : 1. [10 points] Let X be exponential with mean 1 . Find E [ X | X > 1]. Solution: This is a starred problem in the textbook, see solution at page 741. 1
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2. [10 points] a) Write down the probability density function of a normal distributed random variable X having parameter μ and σ 2 . b) Calculate the moment generating function of X . c) How is X + Y distributed, when X, Y are independent normal random variables? Why? Solution: These are textbook examples. 2
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3. [20 points] Let X 1 , X 2 , · · · , X n be independent and identically distributed continuous random variables with probability distribution F and density function F 0 = f . Let X ( i ) de- note the i -th smallest of these random variables, compute the probability density distribution of X ( i ) . Solution: This is Example 2.37 in the textbook. 3
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4. [20 points] Two points X, Y independently and uniformly distributed within the unit interval [0 , 1].
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