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**Unformatted text preview: **Tenth Homework Set — Solutions Chapter 5 Problem 37 Let X be uniformly distributed over (- 1 , 1). (a) P braceleftbig | X | > 1 2 bracerightbig = P braceleftbig X > 1 2 bracerightbig + P braceleftbig X <- 1 2 bracerightbig = 1 2 (b) Let Y = | X | . If y ∈ (0 , 1), then F Y ( y ) = P { Y ≤ y } = P {- y ≤ Y ≤ y } = y , so that f Y ( y ) = braceleftBigg 1 < y < 1 otherwise Problem 39 Let X be exponential with λ = 1, and let Y = log X . Then F Y ( y ) = P { Y ≤ y } = P { log X ≤ y } = P { X ≤ e y } = 1- e- e y , so that f Y ( y ) = e y- e y . Problem 40 Let X be uniform on (0 , 1), and Y = e X . Then, for 1 < y < e , F Y ( y ) = P { Y ≤ y } = P braceleftbig e X ≤ y bracerightbig = P { X ≤ log Y } = log Y , so that f Y ( y ) = braceleftBigg 1 y 1 < y < e otherwise Problem 41 For any r ∈ (- A, A ), we have F R r = P { R ≤ r } = P { A sin θ ≤ r } = P braceleftbig θ ≤ arcsin r A bracerightbig = 1 π arcsin r A , so that f R ( r ) = braceleftBigg 1 π √ A 2- r...

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