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Hw7sol - ISyE 2027B Probability with Applications Fall 2010...

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Unformatted text preview: ISyE 2027B Probability with Applications Fall 2010 Homework 7 Solutions 1. For a < 1, F X ( a ) = 0; for a ≥ 1, F X ( a ) = ∫ a 1 α x α +1 dx = 1- a- α Thus F Y ( y ) = P ( Y ≤ y ) = P (ln( X ) ≤ y ) = P ( X ≤ e y ) = F X ( e y ) . For y < 0 ( e y < 1), F Y ( y ) = 0; for y ≥ 0 ( e y ≥ 1), F Y ( y ) = 1- e- αy . Y has exponential distribution with parameter α . 2. F Y ( y ) = P ( Y ≤ y ) = P (- X ≤ y ) = P ( X ≥ y ) = 1- P ( X ≤ y ) = 1- F X (- y ) Take derivative on both sides, f Y ( y ) = f X (- y ). 3. Since we know f Z ( z ) = nz n- 1 and f V ( v ) = n (1- v ) n- 1 for z,v ∈ [0 , 1]. (a) Let n = 2, f Z ( z ) = 2 z and f V ( v ) = 2(1- v ), where Z = max { X 1 ,X 2 } and V = min { X 1 ,X 2 } . E [ Z ] = ∫ 1 2 z 2 dz = 2 3 E [ V ] = ∫ 1 2(1- v ) vdv = 1 3 (b) Similarly, when Z = max { X 1 ,X 2 ,...,X n } and V = min { X 1 ,X 2 ,...,X n } . E [ Z ] = ∫ 1 znz n- 1 dz = ∫ 1 nz n dz = n n + 1 E [ V ] = ∫ 1 vn (1- v ) n- 1 dv = ∫ 1 n (1- u ) u n- 1...
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Hw7sol - ISyE 2027B Probability with Applications Fall 2010...

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