Introduction to Probability and Statistics for Engineers and Scientists

5.8Distributions Arising from the Normal 189

and 5.7.4 that the mean and variance of a random variableXhaving this distribution is

E[X]=n, Var(X)= 2 n

5.8.2 Thet-Distribution

IfZandχn^2 are independent random variables, withZhaving a standard normal distribu-
tion andχn^2 having a chi-square distribution withndegrees of freedom, then the random
variableTndefined by

Tn=

Z

√

χn^2 /n

is said to have at-distribution with n degrees of freedom. A graph of the density function of
Tnis given in Figure 5.14 forn=1, 5, and 10.
Like the standard normal density, thet-density is symmetric about zero. In addition, asn
becomes larger, it becomes more and more like a standard normal density. To understand
why, recall thatχn^2 can be expressed as the sum of the squares ofnstandard normals,
and so

χn^2

n

=

Z 12 +···+Zn^2

n

whereZ 1 ,...,Znare independent standard normal random variables. It now follows from
the weak law of large numbers that, for largen,χn^2 /nwill, with probability close to 1,
be approximately equal toE[Zi^2 ]=1. Hence, fornlarge,Tn =Z/

√
χn^2 /nwill have
approximately the same distribution asZ.
Figure 5.15 shows a graph of thet-density function with 5 degrees of freedom
compared with the standard normal density. Notice that thet-density has thicker “tails,”
indicating greater variability, than does the normal density.

n = 10

n = 5

n = 1

FIGURE 5.14 Density function of Tn.