1. **Population**: The entire set of individuals or items of interest in a study. Mathematically, it can be represented as a set $\mathcal{P} = \{x_1, x_2, \ldots, x_N\}$ where $N$ is the population size. 2. **Sample**: A subset of the population selected for analysis. If $n$ is the sample size, then the sample is $\mathcal{S} = \{x_{i_1}, x_{i_2}, \ldots, x_{i_n}\}$ with $\mathcal{S} \subseteq \mathcal{P}$. 3. **Sampling**: The process of selecting a sample from the population, often denoted as a function or rule $f: \mathcal{P} \to \mathcal{S}$. 4. **Random Sampling**: A sampling method where each member of the population has an equal probability of being selected. Formally, $P(x_i \in \mathcal{S}) = \frac{n}{N}$ for all $i$. 5. **Statistic**: A function of the sample data used to estimate population parameters, e.g., sample mean $\bar{X} = \frac{1}{n} \sum_{i=1}^n X_i$. 6. **Random Sample**: A sample consisting of independent and identically distributed (i.i.d.) random variables $X_1, X_2, \ldots, X_n$ drawn from the population distribution. 7. **Estimator (Point estimator)**: A rule or function $\hat{\theta} = g(X_1, X_2, \ldots, X_n)$ used to estimate an unknown parameter $\theta$. 8. **Biased Estimator**: An estimator whose expected value does not equal the true parameter, i.e., $E[\hat{\theta}] \neq \theta$. 9. **Unbiased Estimator**: An estimator with expected value equal to the parameter, $E[\hat{\theta}] = \theta$. 10. **Consistent Estimator**: An estimator $\hat{\theta}_n$ such that $\hat{\theta}_n \xrightarrow{p} \theta$ as $n \to \infty$ (converges in probability). 11. **Confidence Interval**: An interval $[L(X), U(X)]$ computed from sample data such that $P(L(X) \leq \theta \leq U(X)) = 1 - \alpha$ for confidence level $1-\alpha$. 12. **Null Hypothesis**: A statement $H_0$ about the population parameter, e.g., $H_0: \theta = \theta_0$, to be tested. 13. **Test Statistic**: A function $T(X)$ of the sample used to decide whether to reject $H_0$. 14. **Critical Region**: The set of values of $T(X)$ for which $H_0$ is rejected, denoted $C$. 15. **Decision Rule**: A rule that rejects $H_0$ if $T(X) \in C$ and fails to reject otherwise. 16. **Type I Error**: Rejecting $H_0$ when it is true; probability $\alpha = P(T(X) \in C | H_0)$. 17. **Type II Error**: Failing to reject $H_0$ when it is false; probability $\beta = P(T(X) \notin C | H_1)$. 18. **Power Function**: The function $\pi(\theta) = P(T(X) \in C | \theta)$ giving the probability of rejecting $H_0$ for each $\theta$. These definitions provide a mathematical foundation for parameter estimation and hypothesis testing.