设随机变量X的密度函数为f(x)，若对于常数c，有f(c+x)=f(c-x)，x＞0且E(X)存在，证明：E(X)=c

Question

如题所述

Answer 1

【答案】：EX=在区间(-无穷大,+无穷大)积分xf(x)dx,换元:u=x-c,du=dx 有
=在区间(-无穷大,+无穷大)积分(u+c)f(u+c)du,
=在区间(-无穷大,+无穷大)积分[uf(u+c) +cf(u+c)]du
容易知道:f(u)=uf(u+c)为奇函数:g(-u)=-uf(-u+c)= -uf(u+c)= -g(u) (由假设得)
故在区间(-无穷大,+无穷大)积分uf(u+c)du=0.
故:EX=0+在区间(-无穷大,+无穷大)积分cf(u+c)du=
=c*{在区间(-无穷大,+无穷大)积分f(u+c)du}=c*1=c,
即EX=c,