NO.PZ2020010301000006
问题如下：
Suppose that 10% of fund managers are superstars. Superstars have a 20% chance of beating their benchmark by more than 5% each year(high return), whereas normal fund managers have only a 5% chance of beating their benchmark by more than 5%.
Continue the application of Bayes’ rule to compute the probability that a manager is a superstar after observing two years of “high” returns.
选项：
解释：
Consider the three scenarios: (High, High), (High, Low) and (Low, Low). We are interested in Pr (Star|High, High) using Bayes’ rule, this is equal to
Pr(High, High|Star)Pr(Star) /Pr(High, High).
Stars produce high returns in 20% of years, and so Pr(High, High|Star) = 20% * 20% Pr (Star) is still 10%.
Finally, we need to compute Pr (High, High), which is Pr(High, High|Star) Pr(Star) + Pr(High, High|Normal)Pr(Normal).
This value is 20% * 20% * 10% + 5% * 5% * 90% = 0.625%. Combing these values,
20% * 20% * 10%/0.625%=64%
This is a large increase from the 30% chance after one year.
题中不需要考虑第二年是否仍为superstar这一条件吗？
