 Mathematical statistics

Mathematical statistics studies the mathematical properties of procedures for learning from data: estimation, statistical hypothesis tests and quantification of uncertainty under explicit statistical models. It asks what a method estimates, how its error behaves and which assumptions justify that behavior.

If $X_1,\dots ,X_n$ are independent and identically distributed random variables with expected value $\mu$ and finite variance ${\sigma }^2$, the sample mean $\overline{X}$ satisfies $\mathbb{E}\overline{X}=\mu$ and $\mathrm{Var}(\overline{X})={\sigma }^2/n$. Independence removes cross-covariances in the second calculation. These identities explain why averaging can improve precision, while also exposing a limitation: strongly dependent observations may provide much less information. Theorems about estimators are therefore conditional statements about a sampling model, rather than universal guarantees for every dataset.

Mathematical statistics is a branch of mathematics that focuses on the theory and methodology of statistical analysis. It combines mathematical theories and tools with statistical principles to understand and analyze data. The main components of mathematical statistics include: 1. **Probability Theory**: This provides the foundational framework for making inferences from data. It involves studying random variables, probability distributions, expectations, and various convergence concepts (such as convergence in distribution, probability, and mean).