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题目

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the National Football League playoffs are just about to begin.Because of their great record in the regular season,the Steelers get a bye in the regular season,the Steelers get a bye in the first week of the playoffs.In the second week,they will play the winner of the game between the Ravens and the Patriots.A football expert estimates that the Ravens will beat the Patriots with probability 0.45.This same expert estimates that that if the Steelers play the Ravens,the mean and standard deviation of the point spread(Steelers points minus Ravens points) will be 6.5 and 10.5,whereas if the steelers play the Patriots,the mean and standard deviation of the point spread( Steelers points minus Patriots points) will be 3.5 and 12.5.Find the mean and standard deviation of the point spread ( Steelers points minus their opponent's points) in the Steelers game.

答案

期望值是6.5*0.45+3.5*0.55=4.85.方差是：根据条件方差的计算公式D(X|A)=E(X^2|A)-E(X|A)^2,得E(X^2|A)=D(X|A)+E(X|A)^2=10.5^2+6.5^2同理E(X^2|B)=D(X|B)+E(X|B)^2=3.5^2+12.5^2所以,D(X)=0.45*(6.5^2+10.5^2)+0.55...