An overview of the fundamental concepts and principles of mathematical statistics, including the role of statistics in data analysis, types of data, and key terminologies.
Introduction to Mathematical Statistics
Key concepts, definitions, and tools that form the foundation of statistical inference.
| Aspect | Description |
|---|---|
| Goal | Use probability theory to model data‑generating processes and to draw conclusions (estimation, testing, prediction). |
| Scope | • Probability models (distributions, random variables) • Sampling theory (iid samples, convergence) • Estimation theory (point, interval) • Hypothesis testing • Asymptotic methods (large‑sample approximations) |
| Contrast with Descriptive Statistics | Descriptive statistics summarises observed data; mathematical statistics provides the theoretical justification for those summaries and for decisions based on data. |
| Family | Typical Parameters | Key Properties |
|---|---|---|
| Bernoulli (B(p)) | (p\in(0,1)) | (\mu=p,; \sigma^2=p(1-p)) |
| Binomial (Bin(n,p)) | (n\in\mathbb{N}, p) | Sum of (n) iid Bernoulli |
| Poisson (Pois(\lambda)) | (\lambda>0) | Mean = variance = (\lambda) |
| Normal (N(\mu,\sigma^2)) | (\mu\in\mathbb{R}, \sigma^2>0) | Closed under linear transformations |
| Exponential (Exp(\lambda)) | (\lambda>0) | Memoryless property |
| Gamma (Gamma(\alpha,\beta)) | shape (\alpha), rate (\beta) | Generalises exponential, chi‑square |
| Statistic | Distribution (under normality) | Use |
|---|---|---|
| Sample mean (\bar X) | (N(\mu,\sigma^2/n)) | Point estimation, CI |
| Sample variance (S^2) | (\sigma^2 |