Homework8

# Homework8 - 2 64 = 25 600(c P C&amp;gt 600 = P...

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HOMEWORK PROBLEMS #8 SOLUTIONS TO EVEN PROBLEMS 18. (a) 1 = Z -∞ f ( y ) dy = Z 0 - 1 . 2 dy + Z 1 0 ( . 2 + cy ) dy = . 2 + . 2 + c 2 , so that c = 1 . 2. (b) F ( y ) = Z y -∞ f ( t ) dt = 0 , y ≤ - 1 R y - 1 . 2 dt = . 2( y + 1) , - 1 < y 0 F (0) + R y 0 ( . 2 + 1 . 2 t ) dt = . 2 + . 2 y + . 6 y 2 , 0 < y 1 1 , y > 1 (c) (d) F ( - 1) = 0 ,F (0) = . 2(0 + 1) = . 2 ,F (1) = . 2 + . 2(1) + . 6(1) 2 = 1. (e) P (0 Y . 5) = F ( . 5) - F (0) = . 2 + . 2( . 5) + . 6( . 5) 2 - . 2 = . 45 - . 2 = . 25. (f) P ( Y > . 5 | Y > . 1) = P ( Y > . 5) Y > . 1 = 1 - F ( . 5) 1 - F ( . 1) = 1 - . 45 1 - ( . 2 + . 2( . 1) + . 6( . 1) 2 ) = . 55 . 774 0 . 71. 22. E ( Y ) = Z -∞ yf ( y ) dy = Z 0 - 1 . 2 ydy + Z 1 0 ( . 2 y + 1 . 2 y 2 ) dy = . 1(0 2 - ( - 1) 2 ) + . 1(1 2 - 0 2 ) + . 4(1 3 - 0 3 ) = . 4. E ( Y 2 ) = Z -∞ y 2 f ( y ) dy = Z 0 - 1 . 2 y 2 dy + Z 1 0 ( . 2 y 2 +1 . 2 y 3 ) dy = . 2 3 (0+1)+ . 2 3 (1 - 0)+ . 3(1 - 0) = 13 30 . 43. Var( Y ) = 13 30 - ( 2 5 ) 2 = 41 150 . 27. 28. (a) Note that this is the pdf of a Beta(3 , 5) distribution, so that c = 1 B (3 , 5) = Γ(8) Γ(3)Γ(5) = 105. You can also ﬁnd it by integrating the pdf as in problem 18(a). (b) Again, you can ﬁnd it by integrating like in problem 22, but we know that for a Beta(3 , 5) distribution, E ( Y ) = 3 3+5 = 3 8 . 375. 32. (a) E ( Y ) = Z 4 0 3 64 (4 y 3 - y 4 ) dy = 3 64 ± 4 4 - 4 5 5 ² = 2 . 4. E ( Y 2 ) = Z 4 0 3 64 (4 y 4 - y 5 ) dy = 3 64 ± 4 6 5 - 4 6 6 ² = 6 . 4. Var( Y ) = 6 . 4 - 2 . 4 2 = . 64. (b) E ( C ) = E (200 Y ) = 200 E ( Y ) = 200(2 . 4) = 480. Var( C ) = Var(200 Y ) = 200 2 Var( Y ) = 200

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Unformatted text preview: 2 ( . 64) = 25 , 600. (c) P ( C &amp;gt; 600) = P ( Y &amp;gt; 3) = Z 4 3 3 64 (4 y 2-y 3 ) dy = 3 64 4 3 ( 4 3-3 3 )-1 4 ( 4 4-3 4 ) = 67 256 . 26, so yes, I would expect it to happen more than a quarter of the time. 1 2 SOLUTIONS TO EVEN PROBLEMS 58. (a) (b) P (0 Z 1 . 2) = 0 . 5-. 1151 = . 3849 P (-. 9 Z 0) = . 5-. 1841 = . 3159 (c) (d) P ( . 3 Z 1 . 56) = . 3821-. 0594 = . 3227 P (-. 2 Z . 2) = 1-2( . 4207) = . 1586 (e) P (-1 . 56 Z -. 2) = . 4207-. 0594 = . 3613 62. (a) (b) P ( Z 2 &amp;lt; 1) = P (-1 &amp;lt; Z &amp;lt; 1) = P ( Z 2 &amp;lt; 3 . 84146) = P (-1 . 96 &amp;lt; Z &amp;lt; 1 . 96) = = 1-2( . 1587) = . 6826 = 1-2( . 0250) = . 9500...
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Homework8 - 2 64 = 25 600(c P C&amp;gt 600 = P...

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