Mathematical Methods for Physics and Engineering - Matematica.NET

STATISTICStherefore consider the x i as a set of N random variables. In the most general case,these random variables will be described by some N-dimensional joint probabilitydensity function P (x 1 ,x 2 ,...,x N ). § In other words, an experiment consisting of Nmeasurements is considered as a single random sample from the joint distribution(or population) P (x), where x denotes a point in the N-dimensional data spacehaving coordinates (x 1 ,x 2 ,...,x N ).The situation is simplified considerably if the sample values x i are independent.In this case, the N-dimensional joint distribution P (x) factorises into the productof N one-dimensional distributions,P (x) =P (x 1 )P (x 2 ) ···P (x N ). (31.1)In the general case, each of the one-dimensional distributions P (x i ) may bedifferent. A typical example of this occurs when N independent measurementsare made of some quantity x but the accuracy of the measuring procedure variesbetween measurements.It is often the case, however, that each sample value x i is drawn independentlyfrom the same population. In this case, P (x) is of the form (31.1), but, in addition,P (x i ) has the same form for each value of i. The measurements x 1 ,x 2 ,...,x Nare then said to form a random sample of size N from the one-dimensionalpopulation P (x). This is the most common situation met in practice and, unlessstated otherwise, we will assume from now on that this is the case.31.2 Sample statisticsSuppose we have a set of N measurements x 1 ,x 2 ,...,x N . Any function of thesemeasurements (that contains no unknown parameters) is called a sample statistic,or often simply a statistic. Sample statistics provide a means of characterising thedata. Although the resulting characterisation is inevitably incomplete, it is usefulto be able to describe a set of data in terms of a few pertinent numbers. We nowdiscuss the most commonly used sample statistics.§ In this chapter, we will adopt the common convention that P (x) denotes the particular probabilitydensity function that applies to its argument, x. This obviates the need to use a different letterfor the PDF of each new variable. For example, if X and Y are random variables with differentPDFs, then properly one should denote these distributions by f(x) andg(y), say. In our shorthandnotation, these PDFs are denoted by P (x) andP (y), where it is understood that the functionalform of the PDF may be different in each case.1222