5.2.5hard一级题目发布者: ai-batch题干从同一总体中抽取两个容量分别为 n,mn, mn,m 的样本，样本均值分别为 xˉ1,xˉ2\bar{x}_1, \bar{x}_2xˉ1,xˉ2，样本方差分别为 s12,s22s_1^2, s_2^2s12,s22，将两组样本合并，其均值、方差分别为 xˉ,s2\bar{x}, s^2xˉ,s2，证明： xˉ=nxˉ1+mxˉ2n+m,\bar{x} = \frac{n\bar{x}_1 + m\bar{x}_2}{n+m},xˉ=n+mnxˉ1+mxˉ2, s2=(n−1)s12+(m−1)s22n+m−1+nm(xˉ1−xˉ2)2(n+m)(n+m−1).s^2 = \frac{(n-1)s_1^2 + (m-1)s_2^2}{n+m-1} + \frac{nm(\bar{x}_1 - \bar{x}_2)^2}{(n+m)(n+m-1)}.s2=n+m−1(n−1)s12+(m−1)s22+(n+m)(n+m−1)nm(xˉ1−xˉ2)2.