Monte hall has removed an incorrect option from the

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Unformatted text preview: te Hall has removed an incorrect option from the unchosen doors, so contestants who switch double their chances of winning the prize. 4. Funds A, B, and C can each make $1 or lose $1 with 50% probability. Assume the funds are pairwise symmetric: for any pair of funs i and j, (= 1&= 1) = (= -1&= -1) (= 1&= -1) = (= -1&= 1) The correlation between A and B is , and the correlation between B and C is also . How high does need to be so that the correlation between A and C is necessarily positive? Solution: Let ( ) ( ) ( ) ( ) Then we have: We know: Hence, ( ) ( ) ( ( (( (( (( Hence, ( Similarly, since ( ( ( ) ( ( ( ( ) ) ( )( )( ) )( )( ) )( )( ) )( )( ) ( ) ( )) ) ) )( )( ( ) ) ) ) )( )( ( )( )( ) (( ) ) )( )( ) ( ) ( ) ( ) ) ) ( ( ( ) ) ) ( ) ( ( Therefore, we need ( ) ( Note: if we have two variables: A and B, which could be 0 or 1. Then, A=1 has two possibilities: 1. A=1 and B=1 2. A=1 and B=-1 ( ) ( Hence, ( ) Therefore, ( ) ( ( ( Remember that ( ( Therefore, ( ) ( ( ) ( ) ) ) ( ) ( ) ) ) ) ) ( ) ) ) ( ( ) ( ) ( ) i.e., we need:...
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