Statistics is a science which deals with collection, tabulation , presentation, analysis, and interpretation of the data. Mathematical statistics is the branch of statistics deals in terms of a mathematical point of view based on the theory of probability and some of the concepts of mathematics such as linear algebra and analysis. It includes the derivation of some theorems related to basic probability theory, random variables and probability distributions, and Normal and associated distributions. In particular, mathematical statistics is mainly deals with the basic concepts of statistical theory such as likelihood functions, sufficient statistics, and exponential family of distributions. Furthermore, it clearly derives the theorems and proofs related to the concepts of point estimation, minimum variance unbiased estimate (MVUE), Cramer-Rao inequality, Maximum likelihood estimates and asymptotic theory.
It is important to be noted that mathematical statistics includes advanced distribution theory such as Multivariate distributions, Transformations, Moment Generating Functions (MGF) and inequalities, Some limit theorems, and further discrete distributions such as Negative binomial, Hyper-geometric and Multinomial distributions with relevant statements and proofs. In addition, it shows the derivations for some continuous distributions such as Gamma and Beta functions, Gamma distribution, and Beta distribution. Finally, some important derivations are clearly discussed with proofs for the concept of multivariate normal distribution, Chi-square, t, and F distribution, and Normal theory tests and confidence intervals. On the whole, one can say that Mathematical statistics acts as a fundamental for all the advanced concepts dealt with statistical theory.
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