Sample Mean and Sample Total: Expectation and Variance | 样本均值与样本总和的期望和方差

📚 Sample Mean and Sample Total: Expectation and Variance | 样本均值与样本总和的期望和方差

In statistical inference, we use the sample mean X̄ and the sample total T to summarise data and estimate population parameters. Understanding their expected values and variances is essential for quantifying sampling error and for constructing confidence intervals or hypothesis tests.

在统计推断中，我们常用样本均值 X̄ 和样本总和 T 来汇总数据并估计总体参数。理解它们的期望与方差，是量化抽样误差、构造置信区间或进行假设检验的基础。

1. Setting and Notation | 设定与符号

Before a sample is drawn, the observations are random variables. Suppose X₁, X₂, …, Xₙ are independent and identically distributed (i.i.d.), with common mean E(Xᵢ) = μ and common variance Var(Xᵢ) = σ² for every i. Define the sample mean and the sample total as:

X̄ = (X₁ + X₂ + … + Xₙ)/n, T = X₁ + X₂ + … + Xₙ

在抽取样本之前，每个观测值都是随机变量。设 X₁, X₂, …, Xₙ 独立同分布（i.i.d.），且每个 Xᵢ 都有共同期望 E(Xᵢ) = μ、共同方差 Var(Xᵢ) = σ²。定义样本均值和样本总和如下：

X̄ = (X₁ + X₂ + … + Xₙ)/n，T = X₁ + X₂ + … + Xₙ

Throughout this article, the population is treated as infinite, or sampling is assumed to be with replacement, so that the observations remain independent.

本文假设总体为无限总体，或抽样为有放回抽样，从而保证各观测值相互独立。

2. Expectation of the Sample Mean | 样本均值的期望

Because expectation is linear, the expectation of a sum is the sum of the expectations, and constant factors can be pulled out. Applying this to X̄ gives:

E(X̄) = E((X₁ + X₂ + … + Xₙ)/n) = (1/n)(E(X₁) + E(X₂) + … + E(Xₙ)) = (1/n)(nμ) = μ

So the mean of the sample mean is exactly the population mean μ. This is why X̄ is called an unbiased estimator of μ: on average, over many samples, X̄ neither overestimates nor underestimates the true mean.

由于期望是线性运算，和的期望等于期望之和，常数因子可以提到外侧。对 X̄ 应用这一性质，得到：

E(X̄) = E((X₁ + X₂ + … + Xₙ)/n) = (1/n)(E(X₁) + E(X₂) + … + E(X