Fundamentals of Probability and Statistics for Engineers

The parameter n is generally referred to as the degrees of freedom. The utility of
this distribution arises from the fact that a sum of the squares of n independent
standardized normal random variables has a^2 distribution with n degrees of
freedom; that is, if U 1 ,U 2 ,..., and Un are independent and distributed as
N (0, 1), the sum

has a^2 distribution with n degrees of freedom. One can verify this statement
by determining the characteristic function of each Uj^2 (see Example 5.7, page
132) and using the method of characteristic functions as discussed in Section 4.5
for sums of independent random variables.
Because of this relationship, the^2 distribution is one of our main tools in
the area of statistical inference and hypothesis testing. These applications are
detailed in Chapter 10.

0 2 4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

fX(x)

n = 1

n = 2

n = 4

n = 6

x

Figure 7.12 The^2 distribution for n 1, n 2, n 4, and n 6

220 Fundamentals of Probability and Statistics for Engineers

XˆU 12 ‡U 22 ‡‡Un^2 … 7 : 68 †

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