Deriving the Unbiased Estimator for Population Variance
Consider a random sample \(X_1, X_2, \dots, X_n\) drawn from a population with true variance \(\sigma^2\). The sample variance is defined as \(s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(X_i – \bar{X})^2\). Derive the expected value \(E(s^2)\) and show that \(s^2\) is an unbiased estimator of \(\sigma^2\).
Derivation of Unbiased Sample Variance
Consider a random sample \( x_1, x_2, \ldots, x_n \) drawn from a population with mean \(\mu\) and variance \(\sigma^2\). The sample mean is given by \(\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i\), and the sample variance is defined as \(S^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2\). Derive why the denominator is \(n-1\) by showing that \(E(S^2)=\sigma^2\). In your answer, explain the role of degrees […]