Question

求1道概率论习题的答案详解 题目在补充中

设随机变量X1，X2，....,Xn相互独立，且都在区间（0,1）上服从均匀分布，Z=max（X1，X2，...，Xn)， 求E(Z)，D(Z) 。答案为n/(n+1),n/( (n+2) ×（n+1）^2） 求详解

Answer 1

z>=1时，P{Z<=z}=1,
z<0时，P{Z<=z}=0,
z∈[0,1)时,P{Z<=z}=P{max(X1，X2，...，Xn)<=z}=P{X1<=z∩X2<=z……∩Xn<=z}
=P{X1<=z}P(X2<=z}……P{Xn<=z} ------------ (因为X1，X2，....,Xn相互独立)
=[Fx(z)]^n=z^n 1, z>=1
故Z的分布函数为Fz(z)= z^n, z∈[0,1)
0, z<0

概率密度为fz(z)=nz^(n-1), z∈[0,1)
0, 其他
EZ=∫[0->1] z*fz(z)dz=∫[0->1] nz^ndz=[n/(n+1)]z^(n+1) | [0->1] = n/(n+1)
E(Z^2)=∫[0->1] z*z*fz(z)dz=[n/(n+2)]z^(n+2) | [0->1] = n/(n+2)
DZ=E(Z^2)-(EZ)^2= n/(n+2)-[n/(n+1)]^2=n/( (n+2) ×（n+1）^2）