1. **Stating the problem:** We want to understand the concept of unbiasedness in statistics and see examples. 2. **Definition:** An estimator \(\hat{\theta}\) of a parameter \(\theta\) is **unbiased** if its expected value equals the true parameter, i.e., $$E(\hat{\theta}) = \theta.$$ This means on average, the estimator hits the true value. 3. **Important rule:** Bias is defined as $$\text{Bias}(\hat{\theta}) = E(\hat{\theta}) - \theta.$$ If bias = 0, the estimator is unbiased. 4. **Example 1: Sample mean as estimator of population mean** - Problem: Estimate population mean \(\mu\) using sample mean \(\bar{X}\). - Formula: $$\bar{X} = \frac{1}{n} \sum_{i=1}^n X_i$$ - Expected value: $$E(\bar{X}) = \mu$$ - Conclusion: Sample mean is an unbiased estimator of population mean. 5. **Example 2: Sample variance (with denominator \(n-1\)) as estimator of population variance** - Problem: Estimate population variance \(\sigma^2\) using $$S^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2$$ - Expected value: $$E(S^2) = \sigma^2$$ - Conclusion: This sample variance is unbiased. 6. **Example 3: Sample variance with denominator \(n\) is biased** - Problem: Use $$S_n^2 = \frac{1}{n} \sum_{i=1}^n (X_i - \bar{X})^2$$ - Expected value: $$E(S_n^2) = \frac{n-1}{n} \sigma^2 < \sigma^2$$ - Conclusion: This estimator underestimates variance, so it is biased. 7. **Example 4: Estimator for population proportion** - Problem: Estimate population proportion \(p\) by sample proportion \(\hat{p} = \frac{X}{n}\) where \(X\) is number of successes. - Expected value: $$E(\hat{p}) = p$$ - Conclusion: Sample proportion is unbiased. 8. **Example 5: Estimator for population mean using a biased estimator** - Problem: Use \(\tilde{X} = \frac{1}{n+1} \sum_{i=1}^n X_i\) - Expected value: $$E(\tilde{X}) = \frac{n}{n+1} \mu < \mu$$ - Conclusion: This estimator is biased (underestimates mean). **Summary:** Unbiasedness means the estimator's expected value equals the true parameter. Sample mean and sample variance with denominator \(n-1\) are classic unbiased estimators, while some others can be biased.