400Hw05ans - STAT 400 Fall 2011 Homework #5 (due Friday,...

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STAT 400 Fall 2011 Homework #5 (due Friday, September 30, by 3:00 p.m.) From the textbook: 8th edition ( ) 1. 3.3-8 ( ) 2. 3.3-24 (a),(b) ( )
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3. 3.4-4 ( ) 4. 3.4-8 ( ) 5. 3.5-2 ( )
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6. 3.6-6 ( ) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
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7. Suppose a random variable X has the following probability density function: = otherwise 0 1 1 ) ( C x x x f a) What must the value of C be so that f ( x ) is a probability density function? For f ( x ) to be a probability density function, we must have: 1) f ( x ) 0, 2) ( ) 1 = - dx x f . ( ) C C C dx x dx x f ln ln ln 1 1 1 1 = = = = - - . Therefore, C = e . b) Find P ( X < 2 ). P ( X < 2 ) = ( ) 1 2 1 ln ln 2 1 2 - = = - dx x dx x f = ln 2 . c) Find P ( X < 3 ). P ( X < 3 ) = ( ) 1 1 ln ln 1 3 - = = - e e dx x dx x f = 1 . d) Find μ X = E ( X ). μ X = E ( X ) = ( ) = = - e e dx dx x x dx x f x 1 1 1 1 = e – 1 . e) Find σ X 2 = Var ( X ). E ( X 2 ) = ( ) = = - e e dx x dx x x dx x f x 1 1 2 2 1 = 2 1 2 - e . σ X 2 = Var ( X ) = E ( X 2 ) [ E ( X ) ] 2 = 2 3 4 2 - + - e e 0.242 .
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8. Suppose a random variable X has the following probability density function: f ( x ) = 1 x / 2 , 0 x 2, zero elsewhere a) Find the cumulative distribution function F ( x ) = P ( X x ). b) Find the median of the probability distribution of X. c) Find the probability P( 0.8 X 1.8 ). d) Find μ X = E ( X ). e) Find σ X 2 = Var ( X ). f) Find the moment-generating function of X. a) F ( x ) = 0 for x 0, F ( x ) = 4 2 x x - for 0 x 2, F ( x ) = 1 for x 2. b) median = 2 2 - . c) P(0.8 X 1.8) = 0.35 . d) E(X) = 2 / 3 . e) Var(X) = 2 / 9 . f) Integrating by parts, M( t ) = 2 2 2 2 1 t t t e - - , t 0, M( t ) = 1 , t = 0.
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9. Suppose the length of time X (in hours) it takes for pizza to be delivered by Momma Leona’s Pizza has the probability density function f ( x ) = ( ) < < - . , , , x x x c otherwise 0 1 0 3 2 a) Find the value of c that makes f ( x ) a valid probability density function. Must have
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400Hw05ans - STAT 400 Fall 2011 Homework #5 (due Friday,...

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