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a/probability-theory-and-mathematical-statistics/exercise/5-mathematical-statistics/mathematical-statistics.tex b/probability-theory-and-mathematical-statistics/exercise/5-mathematical-statistics/mathematical-statistics.tex index 980beec..db004ef 100644 --- a/probability-theory-and-mathematical-statistics/exercise/5-mathematical-statistics/mathematical-statistics.tex +++ b/probability-theory-and-mathematical-statistics/exercise/5-mathematical-statistics/mathematical-statistics.tex @@ -36,6 +36,8 @@ \setcounter{page}{1} \section{统计量} +利用期望和方差等数学特征之间的关系进行计算统计量,往往以$\sum\limits_{i=1}^nX_i$或类似的形式。 + \textbf{例题:}已知总体$X$的期望为$EX=0$,方差$DX=\sigma^2$。从总体抽取容量为$n$的简单随机样本,其均值和方差分别为$\overline{X}$,$S^2$。记$S_k^2=\dfrac{n}{k}\overline{X}^2+\dfrac{1}{k}S^2$($k=1,2,3,4$),则()。 $A.E(S_1^2)=\sigma^2$\qquad$B.E(S_2^2)=\sigma^2$ @@ -44,6 +46,12 @@ $C.E(S_3^2)=\sigma^2$\qquad$D.E(S_4^2)=\sigma^2$ 解:$E(S_k^2)=E\left(\dfrac{n}{k}\overline{X}^2+\dfrac{1}{k}S^2\right)=\dfrac{n}{k}E\overline{X}^2+\dfrac{1}{k}E(S^2)=\dfrac{n}{k}((E\overline{X})^2+D\overline{X})+\dfrac{1}{k}E(S^2)=\dfrac{n}{k}\left(0+\dfrac{\sigma^2}{n}\right)+\dfrac{1}{k}\sigma^2=\dfrac{2\sigma^2}{k}$,$\therefore k=2$。 +\textbf{例题:}设$X_i$为来自总体$E(\lambda)$($\lambda>0$)的简单随机样本,记统计量$T=\dfrac{1}{n}\sum\limits_{i=1}^nX_i^2$,求$ET$。 + +解:$ET=E\dfrac{1}{n}\sum\limits_{i=1}^nX_i^2=\dfrac{1}{n}\sum\limits_{i=1}^nEX_i^2=\dfrac{1}{n}\sum\limits_{i=1}^n(DX_i+E^2X_i)=\dfrac{1}{n}\sum\limits_{i=1}^n\left(\dfrac{1}{\lambda^2}+\dfrac{1}{\lambda^2}\right)\\=\dfrac{1}{n}\cdot\dfrac{2n}{\lambda^2}=\dfrac{2}{\lambda^2}$。 + +\textbf{例题:}设$X_i$为来自总体$X$的简单随机样本,而$X\sim B\left(1,\dfrac{1}{2}\right)$。记$\overline{X}=\dfrac{1}{n}\sum\limits_{i=1}^nX_i$,求$P\left\{\overline{X}=\dfrac{k}{n}\right\}$。($0\leqslant k\leqslant n$) + \section{三大分布} \subsection{\texorpdfstring{$\chi^2$分布}{}}