谈谈阿里达摩院数学竞赛压轴题的解法

### According to Elementary mathematics，we can get

$|\xi - Q_i| ^ + = \dfrac{ |\xi - Q_i| |+ (\xi - Q_i) }{2}$

### and then Do expectations , utilize **Cauchy-Schwarz** inequality, we can easily get:

$E |\xi - Q_i| <= | E ( \xi - Q_i )^2 | ^ \dfrac{1}{2} = \sqrt{\sigma_i^2 + (Q_i - u_i^2)}$ ,

$E |\xi - Q_i| ^+ <= \dfrac{ [\sigma_i^2 + (Q_i - u_i)]^2 - (Q_i - u_i) }{2}$

### thus the upper bound of target function is determined. Besides, this upper bound can be sampled from ### a Bernoulli distribution.

### It has weights assign to :

### and at the same time, we can get similar formula

### according to the above result, we can easily transfer this problem as minimizing $Q ^ *$

### obviously, the above proble is convex , it's direct to utilize lagrange multiplier to solve.

### and the inner -layer minimization solution is

### the reason why Qi is like above appearance is based on $Q_i$ 's non-negative property.

### Thus we can easily get optimized solutions from one step optimazition conditions.

### That's all, Thank you. I really appreciate the help of my greatest company , Prof. Xu.

### It was Pro. Zhao Xu who I put all my heart and passion into the research of Probability Theory and ### Mathematical Statistics.