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Lecture 22 _arvind - Lecture 22 Independent random...

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Lecture # 22
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Independent random variables Ex Store A and B, which belong to the same owner, are located in 2 different towns. If the density function of the weekly profit of each store, in thousands of rupees, is given by and the profit of one store is independent of the other, what is the probability that next week one store makes at least Rs.500 more than the other store? < < = otherwise, 0 3 x 1 if 4 / ) ( x x f
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Sol: X Y Let X and Y denote next week's profits of A and B, respectively. The desired probability is P( X Y 1/2) P( Y X 1/2) 2P( X Y 1/2). x / 4 if 1 x 3 y / 4 if 1 y 3 f (x) and f (y) 0 otherwise, 0 ot + + + = + < < < < = = herwise. xy /16 if 1 x 3, 1 y 3 So f (x, y) 0 otherwise. < < < < =
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. 54 . 0 1024 549 ) 4 3 ( 16 1 ] 1 ) 2 / 1 [( 16 1 ] 2 [ 8 1 ) 16 ( 2 }) 2 / 1 1 , 3 2 / 3 : ) , {( ) (( 2 ) 2 1 ( 2 3 2 / 3 2 3 3 2 / 3 2 2 / 1 1 3 2 / 3 2 3 2 / 3 2 / 1 1 = - - = - - = = = - < < < < = + - - dx x x x dx x x dx xy dx dy xy x y x y x X,Y P / Y X P x x
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Expectation and Covariance Def: Let (X,Y) be a 2-D r.v with joint density f XY . Let H(X,Y) be a r.v. The expected vale of H(X,Y), denoted by E[H(X,Y)] is given by: ∑∑ = x all y all XY f ) y , x ( H )] Y , X ( H [ E . 1 provided ∑∑ x all y all XY ) y , x ( f | ) y , x ( H | exists.
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XY 2. E[H(X,Y)] H(x, y) f dydx -∞ -∞ = ∫ ∫ provided ∫ ∫ - - dx dy ) y , x ( f | ) y , x ( H | XY exists for (X,Y) continuous
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Ex. Let X and Y have joint density function Find E(X 2 +Y 2 ) < < < < + = otherwise. 0
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