ORIGIN

Mathematical Statistics Concepts

Probability and Statistics 1 mins294 words

第六章 数理统计的基本概念

一、随机样本

(1)总体与个体

​ a. 一批灯泡的全体组成一个总体,其中每一个灯泡都是一个个体。

​ b. 一个随机变量X或其相应的分布函数$F(x)$成为一个总体。

(2)样本与样本值

​ a. 样本容量$n$,样本值是每个具体的值

​ b. 简单随机抽样:机会均等(代表性),个体相互独立(独立性)

         c. 简单随机样本$(X_n)$,简称样本,观测值称样本值

(3)定理

​ a. 若总体X的分布函数为F(x),则样本$X_{1}, X_{2}, \cdots, X_{n}$的联合分布函数为
$$
F\left(x_{1}, x_{2}, \cdots, x_{n}\right)=\prod_{i=1}^{n} F\left(x_{i}\right)
$$
​ b. 若总体X的概率密度为f(x),则样本$X_{1}, X_{2}, \cdots, X_{n}$的联合概率密度为
$$
f\left(x_{1}, x_{2}, \cdots, x_{n}\right)=\prod_{i=1}^{n} f\left(x_{i}\right)
$$
​ c. 若总体的分布律为p(x),则样本$X_{1}, X_{2}, \cdots, X_{n}$的联合分布律为
$$
p\left(x_{1}, x_{2}, \cdots, x_{n}\right)=\prod_{i=1}^{n} p\left(x_{i}\right)
$$

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