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  "body": "\n## Definition\n\nMathematical statistics is the study of [[statistical inference|FXCQWFY5RV]] with the methods of [[probability theory|0RWE7CPJ32]].\nIts setting is a [[statistical model|JZ6R7J36K5]] $(S, \\mathcal{P})$.\nThe data $x \\in S$ are a value of a random element $X$ whose law $P$ is an unknown member of $\\mathcal{P}$.\nIts objects are [[statistics|036756NFD6]] $T = g(X)$ and the procedures built from them.\nIt judges each procedure by the laws that the procedure has under the members of $\\mathcal{P}$.\n\n## Remarks\n\nExample: in the model of $n$ independent $\\mathcal{N}(\\mu, 1)$ observations with $\\mu \\in \\R$, the sample mean $\\bar{X}_n$ has law $\\mathcal{N}(\\mu, 1/n)$ for every $\\mu$.\nThis law shows that $\\bar{X}_n$ is unbiased for $\\mu$ with variance $1/n$, a typical result of mathematical statistics.\nNon-example: drawing a bar chart of one data set is descriptive statistics; it uses no model $\\mathcal{P}$ and makes no claim about other possible data.\nThe main branches are point estimation, hypothesis testing, confidence sets and decision theory, each in a parametric or a nonparametric form.\n",
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