129_Prob and Stat for Eng Soln_Probability and Statistics for Engineering and the Sciences 6TH ED

129_Prob and Stat for Eng Soln_Probability and Statistics for Engineering and the Sciences 6TH ED

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CHAPTER 4 Section 4.1 1. a. P(x 1) = ] 25 . ) ( 1 0 2 4 1 1 0 2 1 1 = = = x xdx dx x f b. P(.5 X 1.5) = ] 5 . 5 . 1 5 . 2 4 1 5 . 1 5 . 2 1 = = x xdx c. P(x > 1.5) = ] 438 . ) ( 16 7 2 5 . 1 2 4 1 2 5 . 1 2 1 5 . 1 = = = x xdx dx x f 2. F(x) = 10 1 for –5 x 5, and = 0 otherwise a. P(X < 0) = 5 . 0 5 10 1 = dx b. P(-2.5 < X < 2.5) = 5 . 5 . 2 5 . 2 10 1 = dx c. P(-2 X 3) = 5 . 3 2 10 1 = dx d. P( k < X < k + 4) = ] 4 . ] ) 4 [( 10 1 4 10 4 10 1 = + = = + + k k dx k k x k k 3. a. Graph of f(x) = .09375(4 – x 2 ) 3 2 1 0 -1 -2 -3 0.5 0.0 -0.5 x1 f(x1) 129
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Chapter 4: Continuous Random Variables and Probability Distributions b. P(X > 0) = 5 . ) 3 4 ( 09375 . ) 4 ( 09375 . 2 0 3 2 0 2 = = x x dx x c. P(-1 < X < 1) = 6875 . ) 4 ( 09375 . 1 1 2 = dx x d. P(x < -.5 OR x > .5) = 1 – P(-.5 X .5) = 1 - 5 . 5 . 2 ) 4 ( 09375 . dx x = 1 - .3672 = .6328 4. a. ] 1 ) 1 ( 0 ) ; ( 0 2 / 0 2 / 2 2 2 2 2 = = = = θ x x e dx e x dx x f b. P(X 200) = = 200 0 2 / 2 200 2 2 ) ; ( dx e x dx x f x ] 8647 . 1 1353 . 200 0 2 / 2 2 = + = x e P(X < 200) = P(X 200) .8647, since x is continuous. P(X 200) = 1 - P(X 200) .1353 c. P(100 X 200) = = 200 100 ) ; ( dx x f ] 4712 . 200 100 000 , 20 / 2 x e d. For x > 0, P(X x) = = x dy y f ) ; ( x y dx e e y 0 2 / 2 2 2 ] 2 2 2 2 2 / 0 2 / 1 x x y e e = = 5. a. 1 = ( )] () 8 3 3 8 2 0 3 2 0 2 3 ) ( = = = = k k k dx kx dx x f x b. P(0 X 1) = ] 125 . 8 1 1 0 3 8 1 1 0 2 8 3 = = = x dx x c. P(1 X 1.5) = ] 2969 . 1 64 19 3 8 1 3 2 3 8 1 5 . 1 1 3 8 1 5 . 1 1 2 8 3 = = = x dx x d. P(X 1.5) = 1 - ] [ ] 5781 . 1 0 1 64 37 64 27 3 2 3 8 1 5 . 1 0 3 8 1 5 . 1 0 2 8 3 = = = = x dx x 130
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Chapter 4: Continuous Random Variables and Probability Distributions 6. a. 5 4 3 2 1 0 2 1 0 x f(x) b. 1 = = = = 1 1 2 4 2 2 4 3 3 4 ] 1 [ ] ) 3 ( 1 [ k du u k dx x k c. P(X > 3) = 5 . ] ) 3 ( 1 [ 4 3 2 4 3 = dx x by symmetry of the p.d.f d. () 367 . 128 47 ] ) ( 1 [ ] ) 3 ( 1 [ 4 / 1 4 / 1 2 4 3 4 / 13 4 / 11 2 4 3 4 13 4 11 = = = du u dx x X P e. P( |X-3| > .5) = 1 – P( |X-3| .5) = 1 – P( 2.5 X 3.5) = 1 - 313 . 16 5 ] ) ( 1 [ 5 . 5 . 2 4 3 = du u 7. a. f(x) = 10 1 for 25 x 35 and = 0 otherwise b. P(X > 33) = 2 . 35 33 10 1 = dx c. E(X) = 30 20 35 25 2 35 25 10 1 = = x dx x 30 ± 2 is from 28 to 32 minutes: P(28 < X < 32) = ] 4 . 32 28 10 1 32 28 10 1 = = x dx d. P( a x a+2) = 2 . 2 10 1 = + a a dx , since the interval has length 2. 131
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Chapter 4: Continuous Random Variables and Probability Distributions 8. a. 10 5 0 0.5 0.4 0.3 0.2 0.1 0.0 x f(x) b. dy y ydy dy y f ) ( ) ( 10 5 25 1 5 2 5 0 25 1 + = = 10 5 2 5 0 2 50 1 5 2 50 + y y y = 1 2 1 2 1 ) 2 1 2 ( ) 2 4 ( 2 1 = + = + c. P(Y 3) = = ydy 3 0 25 1 18 . 50 9 50 5 0 2 = y d. P(Y 8) = 92 . 25 23 ) ( 8 5 25 1 5 2 5 0 25 1 = + = dy y ydy e. P( 3 Y 8) = P(Y 8) - P(Y < 3) = 74 . 50 37 50 9 50 46 = = f. P(Y < 2 or Y > 6) = 4 . 5 2 ) ( 10 6 25 1 5 2 3 0 25 1 = = + = dy y ydy 9. a. P(X 6) = (after u = x - .5) du e dx e u x = = 5 . 5 0 15 . 6 5 .
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This note was uploaded on 04/30/2008 for the course STAT 412 taught by Professor Shun during the Fall '06 term at University of Michigan.

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129_Prob and Stat for Eng Soln_Probability and Statistics for Engineering and the Sciences 6TH ED

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