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hw2sln

# hw2sln - IEOR 161 Operations Research II University of...

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IEOR 161 Operations Research II University of California, Berkeley Spring 2008 Homework 2 Suggested Solution Chapter 3. 24. (a) Let N=# of flips whose expectation is desired, h=first flip is head, t=first flip is tail. E [ N ] = E [ N | h ] + E [ N | t ] = (1 + 1 1 - p ) p + (1 + 1 p )(1 - p ) = 1 + p 1 - p + 1 - p p (b) Let X=# of heads whose expectation is desired, h=first flip is head, t=first flip is tail. E [ X ] = E [ X | h ] + E [ X | t ] = (1 + 1 1 - p - 1) p + 1(1 - p ) = 1 - p + 1 1 - p (c) Let Y=# of tails whose expectation is desired, h=first flip is head, t=first flip is tail. E [ Y ] = E [ Y | h ] + E [ Y | t ] = 1 p + (1 + 1 p - 1)(1 - p ) = p + 1 - p p (d) Let M=# of heads whose expectation is desired, h=first flip is head, t=first flip is tail. E [ M ] = E [ M | h ] + E [ M | t ] = (1 + 1 + p 1 - p + 1 - p p ) p + (1 + 2 p )(1 - p ) = p 2 1 - p + 2 p 29. (a) Let N=# of shootings whose expectation is desired, h=first shooting hits the target, m=first shooting misses the target. Also let q i =1-p i , i=1,2 for simplicity. μ 1 = E [ N | h ] + E [ N | m ] = ( E [ N | h, h ] p 2 + E [ M | h, m ] q 2 ) p 1 + (1 + μ 2 ) q 1 = (2 p 2 + (2 + μ 1 ) q 2 ) p 1 + (1 + μ 2 ) q 1 Reorganize the equation above, we get μ 1 (1 - p 1 q 2 ) = 1 + p 1 + μ 2 q 1

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