# Introduction to Statistical Inference

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## Abstract

A typical problem in probability theory is of the following form: A sample space and underlying probability function are specified, and we are asked to compute the probability of a given chance event. For example, if *X*_{1}, …, *X*_{ n } are independent Bernoulli random variables with *P*{*X*_{ i } = 1} = 1 − *P*{*X*_{ i } = 0} = *p*, we compute that the probability of the chance event \(\left\{ {\sum\limits_{i = 1}^n {{X_i} = r} } \right\}\), where *r* is an integer with \(0 \le r \le n\), is \(\left( {\begin{array}{*{20}{c}}n\\r\end{array}} \right){p^r}{\left( {1 - p} \right)^{n - r}}\). In a typical problem of statistics it is not a single underlying probability law which is specified, but rather a *class* of laws, any of which may *possibly* be the one which actually governs the chance device or experiment whose outcome we shall observe. We know that the underlying probability law is a member of this class, but we do not know which one it is. The object might then be to determine a “good” way of guessing, on the basis of the outcome of the experiment, which of the *possible* underlying probability laws is the *one* which actually governs the experiment whose outcome we are to observe.

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