- As a sample size increases, the means of samples drawn from a popula-
tion of any distribution will approach the normal distribution. This theo-
rem, when stated rigorously, is known as the central limit theorem (more
details in Chapter 4).
In addition to the normal distribution (Appendix B), topics introduced in
subsequent chapters involve three other continuous distributions:
Thetdistribution (Appendix C)
The chi-square distribution (Appendix D)
TheFdistribution (Appendix E)Thetdistribution is similar to the standard normal distribution in that it is
unimodal, bell-shaped, and symmetrical; extends infinitely in either direction;
and has ameanof zero. This is a family of curves, each indexed by a number
calleddegrees of freedom(df). Given a sample of continuous data, the degrees
of freedom measure the quantity of information available in a data set that can
be used for estimating the population variances^2 (i.e.,n1, the denominator
ofs^2 ). Thetcurves have ‘‘thicker’’ tails than those of the standard normal
curve; their variance is slightly greater than 1 [¼df=ðdf 2 Þ]. However, the
area under each curve is still equal to unity (or 100%). Areas under a curve
from the right tail, shown by the shaded region, are listed in Appendix C; thet
distribution for infinite degrees of freedom is precisely equal to the standard
normal distribution. This equality is readily seen by examining the column
marked, say, ‘‘Area¼.025.’’ The last row (infinite df) shows a value of 1.96,
which can be verified using Appendix B.
Unlike the normal andtdistributions, the chi-square andFdistributions are
concerned with nonnegative attributes and will be used only for certain ‘‘tests’’
in Chapter 6 (chi-square distribution) and Chapter 7 (Fdistribution). Similar to
the case of thet distribution, the formulas for the probability distribution
functions of the chi-square andFdistributions are rather complex mathemati-
cally and are not presented here. Each chi-square distribution is indexed by a
number called thedegrees of freedom r. We refer to it as the chi-square distri-
bution withrdegrees of freedom; its mean and variance arerand 2r, respec-
tively. AnFdistribution is indexed by 2 degrees of freedomðm;nÞ.
3.4 PROBABILITY MODELS FOR DISCRETE DATA
Again, a class of measurements or a characteristic on which individual obser-
vations or measurements are made is called avariable. If values of a variable
may lie at only a few isolated points, we have adiscrete variable; examples
include race, gender, or some sort of artificial grading. Topics introduced in
subsequent chapters involve two of these discrete distributions: the binomial
distribution and the Poisson distribution.
132 PROBABILITY AND PROBABILITY MODELS