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746 Chapter 17 The Theory of Linear Regression with One Regressor

a. E(Yi2) = m2 + s2
b. Y ¡p μ

c. 1 n Y2i ¡p m2 + s2
n
ia= 1

d. 1 n (Yi - Y)2 = 1 n Yi2 - Y2
n n
a a

i=1 i=1

e. 1 n (Yi - Y)2 ¡p s2
n
ia= 1

f. s2 = n 1 n - Y)2 ¡p s2
-
1 ia= 1(Yi

17.15 Z is distributed N (0,1), W is distributed x2n, and V is distributed x2m. Show,
as n S ∞ and m is fixed, that:

a. W>n ¡p 1.

b. Z ¡d N(0,1). Use the result to explain why the t∞ distribution is
1W > n

the same as the standard normal distribution.

c. V>m ¡d xm2 >m. Use the result to explain why the Fm,∞ distribution is
W>n
the same as the x2m>m distribution.

A p p e n d i x

17.1 The Normal and Related Distributions and

Moments of Continuous Random Variables

This appendix defines and discusses the normal and related distributions. The definitions
of the chi-squared, F, and Student t distributions, given in Section 2.4, are restated here for
convenient reference. We begin by presenting definitions of probabilities and moments
involving continuous random variables.

Probabilities and Moments of Continuous
Random Variables

As discussed in Section 2.1, if Y is a continuous random variable, then its probability is
summarized by its probability density function (p.d.f.). The probability that Y falls between
two values is the area under its p.d.f. between those two values. Because Y is continuous,
however, the mathematical expressions for its probabilities involve integrals rather than
the summations that are appropriate for discrete random variables.