HW6 ans - STAT 400 Spring 2011 Homework#6(due Friday March...

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STAT 400 Spring 2011 Homework #6 (due Friday, March 4, by 3:00 p.m.) 1. 3.3-2 (a) , 3.3-4 (a) ( , )
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2. 3.3-2 (b) , 3.3-4 (b) ( , )
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3. 3.3-2 (c) , 3.3-4 (c) ( , )
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4. 3.3-8 ( ) 5. 3.3-24 (a),(b) ( )
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6. 3.4-4 ( ) 7. 3.4-8 ( )
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8. 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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9. Let X be a continuous random variable with the probability density function f ( x ) = k x 2 , 0 x 1, f ( x ) = 0, otherwise. a) What must the value of k be so that f ( x ) is a probability density function? 1) f ( x ) 0, 2) ( ) 1 d = - x x f . ( ) = = = - 1 0 2 1 0 2 d d d 1 x x k x x k x x f 3 3 1 0 1 3 3 k k x k = = = . k = 3 . b) Find the probability P( 0.4 X 0.8 ). P( 0.4 X 0.8 ) = ( ) 8 . 0 4 . 0 d x x f = 4 . 0 8 . 0 d 3 3 8 . 0 4 . 0 2 x x x = = 0.8 3 – 0.4 3 = 0.448 . c) Find the median of the distribution of X. Need
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HW6 ans - STAT 400 Spring 2011 Homework#6(due Friday March...

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