Unformatted text preview: on a pair of lotteries p and q , one that will be not only sufFcient, but also necessary for p to Frstorder stochastically dominate q . ±or any lottery p and a number x , deFne G ( p , x ) = ∑ z ≥ x p ( z ) (the probability that the lottery p yields a prize at least as high as x ). Denote by F ( p , x ) the cumulative distribution function of p , that is, F ( p , x ) = Probability { z  z < x } . Claim: pD 1 q iff for all x , G ( p , x ) ≥ G ( q , x ) (alternatively, pD 1 q iff for all x , F ( p , x ) ≤ F ( q , x ) ). (See Fg. 9.1.) Proof: Let x < x 1 < x 2 < . . . < x K be the prizes in the union of the supports of p and q . ±irst, note the following alternative expression for Eu ( p ) : Eu ( p ) = X k ≥ p ( x k ) u ( x k ) = u ( x ) + X k ≥ 1 G ( p , x k )( u ( x k ) − u ( x k − 1 ))....
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 Fall '10
 aswa
 Probability, Probability theory, Cumulative distribution function

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