【等价转换】—— min/max 的转换与互相转换

0. min 与 max 的转换

{max(X,Y)=X+Y−min(X,Y)min(X,Y)=X+Y−max(X,Y)min(X,Y)+max(X,Y)=X+Y" role="presentation">⎧⎩⎨⎪⎪max(X,Y)=X+Y−min(X,Y)min(X,Y)=X+Y−max(X,Y)min(X,Y)+max(X,Y)=X+Y{max(X,Y)=X+Y−min(X,Y)min(X,Y)=X+Y−max(X,Y)min(X,Y)+max(X,Y)=X+Y

1. max(X,Y)=n" role="presentation">max(X,Y)=nmax(X,Y)=n 的概率求解

将 max(X,Y)" role="presentation">max(X,Y)max(X,Y) 或 min(X,Y)" role="presentation">min(X,Y)min(X,Y) 首先转换为联合概率的形式，进一步通过如变量间的独立性简化计算：

max(X,Y)=n" role="presentation">max(X,Y)=nmax(X,Y)=n 可拆分成如下三项：

• X=n,Y&lt;n" role="presentation">X=n,Y<nX=n,Y<n
• X&lt;n,Y=n" role="presentation">X<n,Y=nX<n,Y=n
• X=n,Y=n" role="presentation">X=n,Y=nX=n,Y=n

当然可将第三项合并到前两项的任意一项中。

则其：

Pr(max(X,Y)=n)=Pr(X&lt;n,Y=n)+Pr(X=n,Y&lt;n)+Pr(X=n,Y=n)=Pr(X&lt;n,Y=n)+Pr(X=n,Y≤n)" role="presentation">Pr(max(X,Y)=n)=Pr(X<n,Y=n)+Pr(X=n,Y<n)+Pr(X=n,Y=n)=Pr(X<n,Y=n)+Pr(X=n,Y≤n)Pr(max(X,Y)=n)=Pr(X<n,Y=n)+Pr(X=n,Y<n)+Pr(X=n,Y=n)=Pr(X<n,Y=n)+Pr(X=n,Y≤n)

同理可将 min(X,Y)=n" role="presentation">min(X,Y)=nmin(X,Y)=n 拆分为如下三项：

• X&gt;n,Y=n" role="presentation">X>n,Y=nX>n,Y=n
• X=n,Y&gt;n" role="presentation">X=n,Y>nX=n,Y>n
• X=n,Y=n" role="presentation">X=n,Y=nX=n,Y=n