• A statistical technique used to provide a single best guess of an unknown population parameter based on sample data.
• Common estimators:sample mean, sample variance, sample proportion
• Notation:
  • Population parameter: \(\theta\) (e.g., \(\mu\), \(\sigma^2\))
    
  • Point estimate: \(\hat{\theta}\)
• Desirable properties of estimators:
  • Unbiasedness: \(E[\hat{\theta}] = \theta\)
  • Consistency: \(\hat{\theta} \to \theta\) as \(n \to \infty\)
  • Efficiency: lowest variance among all unbiased estimators

• Sample Mean (Estimator of \(\mu\)):
  \[ \hat{\mu} = \bar{X} = \frac{1}{n} \sum_{i=1}^{n} X_i \]

• Sample Variance (Estimator of \(\sigma^2\)):
  \[ \hat{\sigma}^2 = \frac{1}{n - 1} \sum_{i=1}^{n} (X_i - \bar{X})^2 \]

• Sample Proportion (Estimator of \(p\)):
  \[ \hat{p} = \frac{x}{n} \]

• Mean Squared Error (MSE):
  \[ \text{MSE}(\hat{\theta}) = \text{Var}(\hat{\theta}) + [\text{Bias}(\hat{\theta})]^2 \]

• Likelihood Function (used in MLE):
  \[ L(\theta) = \prod_{i=1}^{n} f(X_i; \theta) \]

set.seed(123)
pop = rnorm(10000, mean = 50, sd = 10)
sample_means = replicate(1000, mean(sample(pop, 30)))

ggplot(data.frame(x = sample_means), aes(x)) +
  geom_histogram(binwidth = 0.5, fill = "lightblue", color = "black") +
  geom_vline(xintercept = mean(pop), color = "red", linetype = "dashed") +
  labs(title = "Sampling Distribution of Sample Mean",
       x = "Sample Mean", y = "Frequency") +
  theme_minimal()