Hw5 - X . (c) Find P ( X > Y ). 4. X and Y have joint...

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SIEO 3600 (IEOR Majors) Assignment #5 Introduction to Probability and Statistics February 17, 2010 Assignment #5 – due February 23th, 2009 1. X is a continuous random variable with probability density function f X ( x ) = ± 1 / 2 , if 2 < x < 4; 0 , otherwise . (a) Compute P ( X < 3). (b) Compute R 0 xf ( x ) dx and R 0 x 2 f ( x ) dx . (c) Show that the cumulative distribution function F X ( x ) = 0 , if x 2; ( x - 2) / 2 , if 2 < x < 4; 1 , if x 4 . (d) Let ¯ F X ( x ) 1 - F X ( x ), show that ¯ F X ( x ) = 1 , if x 2; (4 - x ) / 2 , if 2 < x < 4; 0 , if x 4 . (e) Graph both F X ( x ) and ¯ F X ( x ). (f) Show that R 0 xf ( x ) dx = R 0 ¯ F ( x ) dx . 2. X and Y are two continuous random variables with joint probability density function f X,Y ( x,y ) = ± 10 xy 2 , 0 < x < y < 1; 0 , otherwise . (a) Confirm that f really is a probability density, that is, confirm that R -∞ R -∞ f X,Y ( x,y ) dxdy = 1. (b) Let rigion A = { ( x,y ) : x > 0 . 5 ,y > 0 . 5 } . Compute P (( X,Y ) A ) = P ( X > 0 . 5 ,Y > 0 . 5). (c) Find the marginal density functions f X ( x ) and f Y ( y ) of X and Y . 3. The joint probability density function of X and Y is given by f X,Y ( x,y ) = 6 7 ² x 2 + xy 2 ³ , 0 < x < 1 , 0 < y < 2 (a) Verify that this is indeed a joint probability density function. (b) Compute the probability density function of
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Unformatted text preview: X . (c) Find P ( X > Y ). 4. X and Y have joint probability density function f X,Y ( x,y ) = ± x (1 + 3 y 2 ) / 4 , < x < 2 , < y < 1; , otherwise . Are X and Y independent? (e.g., check if f X,Y ( x,y ) = f X ( x ) · f Y ( y ).) 2 SIEO 3600, Assignment #5 5. The joint density of X and Y is given by f X,Y ( x,y ) = ( xe-x-y , x > ,y > , , otherwise. (a) Compute the density of X . (b) Compute the density of Y . (c) Are X and Y independent? 6. Let X 1 ,X 2 ,...,X n be independent random variables, each having density f X ( x ) = ( 1 , ≤ x ≤ 1 , otherwise . Let M = max { X 1 ,X 2 ,...,X n } . Show that the distribution function of M , F M ( · ), is given by F M ( x ) = x n , ≤ x ≤ 1 What is the probability density function of M ?...
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This note was uploaded on 10/03/2011 for the course SIEO W3600 taught by Professor Yunanliu during the Spring '10 term at Columbia.

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Hw5 - X . (c) Find P ( X > Y ). 4. X and Y have joint...

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