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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
大意是，足球季后赛，钢人队第一轮轮空，要和雷文斯和爱国者的胜者比赛。一个专家估计，雷文期战胜爱国者的概率为0.45。这个专家还估计，如果钢人队迎战雷文斯队，分差（钢人队分数减去雷文斯队分数）的均值为6.5，标准差为10.5。如果钢人队迎战爱国者队，分差（钢人队分数减去爱国者队分数）的均值为3.5，标准差为12.5。
请估计钢人队与对手分差的均值和标准差。
我的思考是：
期望值是6.5*0.45+3.5*0.55=4.85。
方差是：10.5*0.45+12.5*0.55=11.6 但没有把握，求助统计好手！
最佳答案
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设：雷文期战胜爱国者=A事件发生, P(A)=0.45, 爱国者胜雷文期=B事件发生, P(B)=0.55. A是B的补集。令X=钢人比赛分差。即已知P(A)=0.45，P(B)=0.55，A是B的补集；E(X|A)=6.5， D(X|A)=10.5^2； E(X|B)=3.5， D(X|B)=12.5^2，这里D(X)表示方差。
求：E(X), sigma(X)（即D(X)的平方根）
我的解法是（水平有限，请指正）：
你的E(X)没错，E(X)=E(X|A)*P(A)+E(X|B)*P(B)=0.45*6.5+0.55*3.5=4.85
你算D(X)似乎不对，应该是：
D( ...
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