Question

一个概率的问题

若随机变量X与Y相互独立且服从[0,1]的均匀分布，则Z=MAX{X,Y}的期望E(Z)=?,方差D(Z)=?

Answer 1

FZ(z)=P{Z<z}=P{X<z,Y<z}=P{X<z}*P{Y<z}=FX(z) * FY(z)
求导得:
fZ(z)=fX(z)*FY(z)+fY(z)*FX(z). (1)
注意到:fX(z)=1 (1<= z <=1)
fX(z)=0 其它
及:
FX(z)=0 (负无穷, 0)
FX(z)=z (0<=z<=1)
FX(z)=1 (1<z<正无穷大)
同理对于fY(z), FY(z)
(1)式化为:
fZ(z)=1*z+1*z =2z (0<=z<1)
fZ(z)=0 其它.
由此:E(Z)=在区间[0,1]上积分2z*z=2/3
E(Z^2)=在区间[0,1]上积分2z*(z^2)=2/4=1/2
D(Z)=E(Z^2)-[E(Z)]^2=1/2-(2/3)^2=1/18.

Answer 2

1/2,18/1