5012--10.23 - Formulas F x,y = ∑ u,v ≤ x,y f x,y f Y y...

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Unformatted text preview: 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.) = 7....
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