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5012--10.23 - Formulas F(x y =(u,v(x,y f(x y fY(y = P(Y = y...

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Formulas: F ( x,y ) = ( u,v ) ( x,y ) f ( x,y ), f Y ( y ) = P ( Y = y ) = P ( X < ,Y = y ) = x f ( x,y ). f (6 , 0) = 1 / 36, f (10 , 2) = 2 / 36, f Y (2) = 8 / 36, F (5 , 1) = ( x,y ) (5 , 1) f ( x,y ) = f (2 , 0) + f (3 , 1) + f (4 , 0) + f (5 , 1) = 1 / 6. f ( x,y ) = 1 / 36 if ( x,y ) = (2 i, 0), i ∈ { 1 , 2 ,..., 6 } 2 / 36 if ( x,y ) ∈ { (3 , 1) , (5 , 1) , (7 , 1) , (9 , 1) , (11 , 1) , (4 , 2) , (6 , 2) , (8 , 2) , (10 , 2) , (5 , 3) , (7 , 3) , (9 , 3) , (6 , 4) , (8 , 4) , (7 , 5) } Definition. E ( X ) = ( E ( X 1 ) ,...,E ( X n )). Example 1 (continued). Compute E (( X,Y )). Sol. Three ways: (1) E ( X ) = x xf X 1 ( x ), (2) E ( X ) = x,y xf X,Y ( x,y ). (3) E ( X ) = E ( F )+ E ( S ), where F is the outcome fo the first tossing, S is the outcome fo the second tossing. Method (1): f X ( x ) = y f ( x,y ): 2 3 4 5 6 7 8 9 10 11 12 1 / 36 2 / 36 3 / 36 4 / 36 5 / 36 6 / 36 5 / 36 4 / 36 3 / 36 2 / 36 1 / 36 x xf X ( x ) = 7. Method (2): E ( X ) = 1 36 [2 6 i =1 i ]+ 2 36 [(3+5+7+9+11)+(4+6+8+10)+(5+7+9)+(6+8)+7] = 7. Method (3): E ( X ) = E ( F ) + E ( S ) = 3 . 5 + 3 . 5 = 7 Q: Can we say E ( Y ) = E ( | F - S | ) = | E ( F ) - E ( S ) | = | 3 . 5 - 3 . 5 | = 0 ? Recall that a r.v. X is a map X : S → R . X ( ω ) ∈ R . Def. Given a sample space S , let X 1 , ..., X n be n r.v.s from S → R , then X = ( X 1 ,...,X n ) is called a random vector (from S to R n ). If X 1 , ..., X n are all discrete random variables, then X is discrete. X
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