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11-11 Joint Distr Random Variables (2)

# 11-11 Joint Distr Random Variables (2) - BTRY 4080 STSCI...

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BTRY 4080 / STSCI 4080 Fall 2010 322 Section 6.2: Independent Random Variables Definition: X and Y are said to be independent random variables if, for any two sets of real numbers A and B , P ( X A,Y B ) = P ( X A ) × P ( Y B ) . Equivalently: for all x,y R , P ( X x,Y y ) = P ( X x ) × P ( Y y ) . or p ( x,y ) = p X ( x ) p Y ( y ) when both X and Y are discrete f ( x,y ) = f X ( x ) f Y ( y ) when both X and Y are continuous

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BTRY 4080 / STSCI 4080 Fall 2010 323 Necessary & sufficient condition for independence (two random variables): Proposition 2.1 The continuous (discrete) random variables X and Y are independent if and only if their joint probability density (mass) function can be expressed as f X,Y ( x,y ) = h ( x ) g ( y ) , −∞ < x,y < for a function h ( · ) that depends only on x and a function g ( · ) that depends only on y . This factorization includes the support set – it must be “rectangular” . . .
BTRY 4080 / STSCI 4080 Fall 2010 324 Example (6.2.2f): (a): Let X and Y have joint pdf given by f ( x,y ) = 6 e 2 x e 3 y , 0 < x,y < . Are X and Y independent? (b): Let X and Y have joint pdf given by f ( x,y ) = 24 xy 0 < x,y < 1 , 0 < x + y < 1 . Are X and Y independent? (c): Let X and Y have joint pdf f ( x,y ) = 3 8 ( x 2 + y 2 ) 1 < x,y < 1 Are X and Y independent?

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BTRY 4080 / STSCI 4080 Fall 2010 325 Example (Slide 317): Let X and Y have joint pmf P ( X = x,Y = y ) = 8 < : p 2 (1 p ) y y = 0 , 1 , 2 ... ; x = 0 , 1 , 2 ... ; x y 0 otherwise The restriction x y implies X and Y cannot be independent. Can also confirm this easily using earlier calculations (Slide 317): P ( X = x ) = p (1 p ) x , x = 0 , 1 , 2 ,... and P ( Y = y ) = ( y + 1) p 2 (1 p ) y , y = 0 , 1 , 2 ,... ; clearly, multiplying these two pmfs does not yield the joint pmf described above.
BTRY 4080 / STSCI 4080 Fall 2010 326 Example (6.2.2c): A man and a woman decide to meet at a certain location. Each person independently arrives at a time uniformly distributed between 12 noon and 1 pm. What is the probability that the first to arrive has to wait longer than 10 minutes? Solution: Let X = # of minutes by which the man arrives later than 12 noon. Y = # of minutes by which the woman arrives later than 12 noon. Then, X and Y are independent (by assumption) and the desired probability equals P ( | X Y | ≥ 10) .

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BTRY 4080 / STSCI 4080 Fall 2010 327 P ( | X Y | ≥ 10) = integraldisplay integraldisplay | x y | > 10 f ( x,y ) dx dy (independence) = integraldisplay integraldisplay | x y | > 10 f X ( x ) f Y ( y ) dx dy (find integration region) = integraldisplay 60 10 integraldisplay x 10 0 1 3600 dy dx bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright x y> 10 y<x 10 + integraldisplay 60 10 integraldisplay y 10 0 1 3600 dx dy bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright ( x y ) > 10 x<y 10 = 25 72 + 25 72 = 25 36 .
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11-11 Joint Distr Random Variables (2) - BTRY 4080 STSCI...

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