Chapter4 - 129 CHAPTER 4 Section 4.1 1. a. P(x 1) = ] 25 ....

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Unformatted text preview: 129 CHAPTER 4 Section 4.1 1. a. P(x 1) = ] 25 . ) ( 1 2 4 1 1 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 . 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 1-1-2-3 0.5 0.0-0.5 x1 f(x1) Chapter 4: Continuous Random Variables and Probability Distributions 130 b. P(X > 0) = 5 . ) 3 4 ( 09375 . ) 4 ( 09375 . 2 3 2 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 ( ) ; ( 2 / 2 / 2 2 2 2 2 =--=-= = -- - q q q q x x e dx e x dx x f b. P(X 200) = --= 200 2 / 2 200 2 2 ) ; ( dx e x dx x f x q q q ] 8647 . 1 1353 . 200 2 / 2 2 = +--=-q 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 q ] 4712 . 200 100 000 , 20 / 2 --x e d. For x > 0, P(X x) = = -x dy y f ) ; ( q -x y dx e e y 2 / 2 2 2 q ] 2 2 2 2 2 / 2 / 1 q q x x y e e---=-= 5. a. 1 = ( 29 ] ( 29 8 3 3 8 2 3 2 2 3 ) ( = = = = -k k k dx kx dx x f x b. P(0 X 1) = ] 125 . 8 1 1 3 8 1 1 2 8 3 = = = x dx x c. P(1 X 1.5) = ] ( 29 ( 29 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 - ] ( 29 [ ] 5781 . 1 1 64 37 64 27 3 2 3 8 1 5 . 1 3 8 1 5 . 1 2 8 3 =-=--= = x dx x Chapter 4: Continuous Random Variables and Probability Distributions 131 6. a. 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. ( 29 367 . 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....
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Chapter4 - 129 CHAPTER 4 Section 4.1 1. a. P(x 1) = ] 25 ....

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