### 4. SAMPLING AND PARAMETER ESTIMATION. THE MAXIMUM LIKELIHOOD METHOD

Sampling is the experimental method by which information can be obtained about the parameters of an unknown distribution. As is well known from the debate over opinion polls, it is important to have a representative and unbiased sample. For the experimentalist, this means not rejecting data because they do not ``look right''. The rejection of data, in fact, is something to be avoided unless there are overpowering reasons for doing so.

Given a data sample, one would then like to have a method for determining the best value of the true parameters from the data. The best value here is that which minimizes the variance between the estimate and the true value. In statistics, this is known as estimation. The estimation problem consists of two parts: (1) determining the best estimate and (2) determining the uncertainty on the estimate. There are a number of different principles which yield formulae for combining data to obtain a best estimate. However, the most widely accepted method and the one most applicable to our purposes is the principle of maximum likelihood. We shall very briefly demonstrate this principle in the following sections in order to give a feeling for how the results are derived. The reader interested in more detail or in some of the other methods should consult some of the standard texts given in the bibliography. Before treating this topic, however, we will first define a few terms.