Fundamentals of Probability and Statistics for Engineers

220 Fundamentals of Probability and Statistics for EngineersThe parameter n is generally referred to as the degrees of freedom. The utility ofthis distribution arises from the fact that a sum of the squares of n independentstandardized normal random variables has a 2 distribution with n degrees offreedom; that is, if U 1 ,U 2 ,..., and U n are independent and distributed asN(0, 1), the sumX ˆ U 2 1 ‡ U2 2 ‡‡U2 n…7:68†has a 2 distribution with n degrees of freedom. One can verify this statementby determining the characteristic function of each Uj2 (see Example 5.7, page132) and using the method of characteristic functions as discussed in Section 4.5for sums of independent random variables.Because of this relationship, the 2 distribution is one of our main tools inthe area of statistical inference and hypothesis testing. These applications aredetailed in Chapter 10.f X (x)0.80.6n = 10.4n = 20.2n = 4n = 60.00 2 4 6 8 10 12xFigure 7.12 The 2 distribution for n ˆ 1, n ˆ 2, n ˆ 4, and n ˆ 6TLFeBOOK