数理统计的2道题设X1,X2...Xn来自正态总体N（0,1）,定义Y1=|X拔|,Y2=加和i=1到n的|Xi|/n,计算EY1,EY2

Question

数理统计的2道题
设X1,X2...Xn来自正态总体N
（0,1）,定义Y1=|X拔|,Y2=加和i=1到n的|Xi|/n,计算EY1,EY2

Answer 1

Xˉ=(X1+X2+...+Xn)/n～N(0,1/n),
EY1=E(|Xˉ|）=∫[-∞,+∞]|x|*(n/2π)^(1/2)*e^(-nx^2/2)dx
=2∫[0,+∞]x*(n/2π)^(1/2)*e^(-nx^2/2)dx
=2∫[0,+∞](1/2nπ)^(1/2)*e^(-nx^2/2)d(nx^2/2)
=2[-(1/2nπ)^(1/2)*e^(-nx^2/2)][0,+∞]
=(2/nπ)^(1/2).
总体X～N(0,1).
EY2=E{∑[i=1,n]|Xi|/n}
=∑[i=1,n]{E|Xi|/n}
=∑[i=1,n]{E|X|/n}
=E|X|
=(2/π)^(1/2).