a 1 2 F a 1 2 F a F a 1 2 Thus a median F is the unique critical point ie

A 1 2 f a 1 2 f a f a 1 2 thus a median f is the

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( a ) = 0 1 - 2 F ( a ) = 1 - 2 F ( a ) = 0 F ( a ) = 1 2 Thus, a * = median F is the unique critical point ( i.e. , satisfier of the first order necessary condition). To be complete, check the second order sufficient condition. Recalling that the first derivative of F exists and is denoted by f , d 2 d a 2 EV F ( a ) = d d a 1 - 2 F ( a ) = - 2 f ( a ) Answer. The expected value of a choice a is EV F ( a ) = -∞ - a - x d F = a -∞ - a - x d F + a - a - x d F = a -∞ - ( a - x ) d F + a - ( x - a ) d F Taking derivatives (in order to optimize with respect to the choice a ) d d a EV F ( a ) = d d a a -∞ - ( a - x ) d F + d d a a - ( x - a ) d F = a -∞ d ( - ( a - x )) d a d F + d a d a ( - ( a - a )) + a d ( - ( x - a )) d a d F + d a d a ( - ( a - a )) = a -∞ d ( x - a ) d a d F + d a d a ( - ( a - a )) + a d ( a - x ) d a d F + d a d a ( - ( a - a )) = a -∞ ( - 1 ) d F + ( 1 )( 0 ) + a ( 1 ) d F + ( 1 )( 0 ) = a -∞ ( - 1 ) d F + a ( 1 ) d F = - F ( a ) + ( 1 - F ( a )) = 1 - 2 F ( a ) 2
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