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MIT2_854F10_ind_ex - = 1 f" And(1 g(2 g(6 = 1 g"...

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Notes for Lecture 3 Chuan Shi Example of Independence A = { i = 2 or 3} ; B = { j = 1 or 5 or 6} . Thus, we have A B {(2,1),(3,1),(2,5),(3,5),(2,6),(3,6)} . = So, we can compute the following: ( ) = 12/36 ; P A = 1/3 P B ( ) = 18/36 = 1/ 2 ; P A = ( ) ( ) ( B ) = 6/36 = 1/ 6 P A P B . We can also demonstrate the independence in the following way. Let prob i ( ) j = f i g j ( ) ( ) Thus,
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( ) = f (2) (1) + f g + + f (2) (6) prob A g (2) (2) " g + f (3) (1) + f (3) (2) + + f g g g " (3) (6) = ( (2) + f (3))( (1) + g (2) + + g (6)) f g " Similarly, we have ( ) = ( (1) + g (5) + g (6))( (1) + f (2) + + f (6)) prob B g f " And ( B ) f g + f (3) (1) + f g + f (3) (5) + f g + f (3) (6) prob A = (2) (1) g (2) (5) g (2) (6) g Note that (1) + f (2) + + f (6) = 1 f " And (1) + g (2) + + g (6) = 1
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Unformatted text preview: = 1 f " And (1) + g (2) + + g (6) = 1 g " Therefore, prob A ( ) = f (2) + f (3) ( ) = g (1) + g (5) + g (6) prob B Thus, ( ) ( ) = f + f (3))( (1) + g (5) + g (6)) prob A prob B ( (2) g (2) (1) + f (3) (1) + f g + f (3) (5) + f g + f (3) (6) = f g g (2) (5) g (2) (6) g ( ∩ B ) = prob A So, A and B are independent. MIT OpenCourseWare http://ocw.mit.edu 2.854 / 2.853 Introduction to Manufacturing Systems Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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  • Fall '10
  • StanleyGershwin
  • Following, MIT OpenCourseWare, Massachusetts Institute of Technology, PROB, OpenCourseWare, Chuan Shi

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