Question

关于一道概率论的问题.急·······

Answer 1

(X,Y)的分布密度函数为f(x,y),

当(x,y)属于D时:f(x,y)= 1/2.

其它情况,f(x,y)= 0.

令Z=min {X,Y}

Z的分布函数为F(z) = P{Z<=z}

则当z<0时,F(z)=0

当0<= z <1时, F(z)=(1/2)*[2-(1-z)*(2-z)] =(1/2)[3z- z^2]

当z<=1时, F(z)= 1.

由此,求得Z的密度函数:g(z):

当0<=z<1时,g(z)=(1/2)(3-2z),

z为其它值时, g(z)= 0.

由此, E(Z) = 在区间[0,1]上积分 zg(z)=(1/2)*(3z-2z^2). = 5/12.