 mathematical statistics is the application of mathematics to statistics, which was originally conceived as the science of the state â the collection and analysis of facts about a country: its economy, land, military, population, and so forth. mathematical techniques which are used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure-theoretic probability theory. ==introduction== statistical science is concerned with the planning of studies, especially with the design of randomized experiments and with the planning of surveys using random sampling. the initial analysis of the data from properly randomized studies often follows the study protocol. of course, the data from a randomized study can be analyzed to consider secondary hypotheses or to suggest new ideas. a secondary analysis of the data from a planned study uses tools from data analysis. data analysis is divided into: * descriptive statistics - the part of statistics that describes data, i.e. summarises the data and their typical properties. * inferential statistics - the part of statistics that draws conclusions from data (using some model for the data): for example, inferential statistics involves selecting a model for the data, checking whether the data fulfill the conditions of a particular model, and with quantifying the involved uncertainty (e.g. using confidence intervals). while the tools of data analysis work best on data from randomized studies, they are also applied to other kinds of data --- for example, from natural experiments and observational studies, in which case the inference is dependent on the model chosen by the statistician, and so subjective.freedman, d.a. (2005) statistical models: theory and practice, cambridge university press. isbn 978-0-521-67105-7 mathematical statistics has been inspired by and has extended many procedures in applied statistics. ==topics== the following are some of the important topics in mathematical statistics:hogg, r. v., a. craig, and j. w. mckean. "intro to mathematical statistics." (2005).larsen, richard j. and marx, morris l. "an introduction to mathematical statistics and its applications" (2012). prentice hall. ===probability distributions===  a [[probability distributi
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