设随机变量X,Y相互独立,且服从同一分布,试证明 P{a<min(X,Y)≤b}=[p(X>a)]2-[p(X>b)]2.
【答案】:因为X与Y独立同分布,故P{a<min(X,Y)≤b}=P{min(X,Y)≤b}P{min(X,Y)≤a}
=1-P{min(X,Y)>b}-[1-P{min(X,Y)>a}]
=P{min(X,Y)>a}-P}min(X,Y)>b}
=P(X>a,Y>a)-P(X>b,Y>b)
=P(X>a)P(Y>a)-P(X>b)P(Y>b)
=[P(X>a)]2-[P(X>b)]2.
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