Let x1, x2, ⋯, xn be independent random variables, each with...

Let $x_{1}, x_{2}, \cdots, x_{n}$ be independent random variables, each with density function $f(x)$ expected value $\mu,$ and variance $\sigma^{2} .$ Define the sample mean by $\bar{x}=\sum_{i=1}^{n} x_{i} .$ Show that $\left.E(\bar{x})=\mu, \text { and } \operatorname{Var}(\bar{x})=\sigma^{2} / n . \text { (See Problems } 5.9,5.13, \text { and } 6.15 .\right)$.

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Key Concepts

Linearity of Expectation

This concept states that the expected value of a sum of random variables is equal to the sum of their individual expected values, regardless of whether the variables are independent or not. It is crucial in proving that the expectation of the sample mean, which is defined as the sum of independently distributed random variables divided by the sample size, equals the common mean ?.

Variance of Independent Random Variables

When random variables are independent, the variance of their sum is the sum of their individual variances. This property is essential for calculating the variance of the sample mean because the scaling factor (1/n) affects the variance by its square, leading to the result that Var(x?) equals ?²/n.

Sample Mean as an Unbiased Estimator

The sample mean is used to estimate the true population mean ?. It is considered an unbiased estimator because its expected value is exactly ?. This concept underlines the reliability of the sample mean in approximating the true mean, and its variance, which decreases with the sample size, reflects the increasing precision of the estimate as more data is used.

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