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exam1_2010 - Statistics 403 Midterm Exam October 26th 2010...

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Statistics 403 Midterm Exam October 26th, 2010 1. Answer the questions given below for the following probability distribution (your an- swers may depend on a and p ): x - a 0 a P ( X = x ) p/ 2 1 - p p/ 2 (a) What is the expected value EX of this distribution? Solution: The expected value is - a · p/ 2 + 0 · (1 - p ) + a · p/ 2 = 0 . (b) What is the variance var( X ) of this distribution? Solution: var( X ) = E ( X - EX ) 2 = a 2 · p/ 2 + 0 2 · (1 - p ) + a 2 · p/ 2 = a 2 p (c) What value of p maximizes the variance of this distribution? Solution: p = 1 (d) What is P ( X 0) for this distribution? Solution: P ( X 0) = P ( X = - a ) + P ( X = 0) = p/ 2 + 1 - p = 1 - p/ 2 (e) What value of p maximizes P ( X 0) for this distribution? Solution: p = 0 1

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(f) What is E [1 / (1 + X 2 )] for this distribution? Solution: ( p/ 2) / (1 + a 2 ) + 1 - p + ( p/ 2) / (1 + a 2 ) = 1 - p + p/ (1 + a 2 ) (g) Suppose the value of a is restricted so a 1. What value of a maximizes the value of E [1 / (1 + X 2 )] for this distribution? Solution: a = 1 2
2. Suppose we are comparing two samples of data, X 1 , . . . , X n , and Y 1 , . . . , Y 2 n (note that one sample is twice as large as the other). Our goal is to learn about the relationship between their population means EX and EY . The two populations have the same variance, which we denote σ 2 . (a) If σ = 1, what sample size is required so that the standard deviation of ¯ X - ¯ Y is less than 1 / 10?

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exam1_2010 - Statistics 403 Midterm Exam October 26th 2010...

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