Page 748 -

17.5 Weighted Least Squares 747

Let fY denote the probability density function of Y. Because probabilities cannot be
negative, fY(y) Ú 0 for all y. The probability that Y falls between a and b (where a < b) is

b

Pr(a … Y … b) = fY(y)dy. (17.32)
La

Because Y must take on some value on the real line, Pr( - ∞ … Y … ∞ ) = 1, which implies
∞
that fY(y)dy = 1.
1-∞
Expected values and moments of continuous random variables, like those of discrete

random variables, are probability-weighted averages of their values, except that summa-

tions [for example, the summation in Equation (2.3)] are replaced by integrals. Accord-

ingly, the expected value of Y is

E(Y) = mY = yfY(y)dy, (17.33)
L

where the range of integration is the set of values for which fY is nonzero. The variance is
the expected value of (Y - mY)2, the rth moment of a random variable is the expected value
of Yr, and the rth central moment is the expected value of (Y - mY)r. Thus

var(Y) = E(Y - mY)2 = (y - mY)2 fY(y)dy, (17.34)
L

E(Y r) = yrfY(y)dy, (17.35)
L

and similarly for the rth central moment, E(Y - mY)r.

The Normal Distribution

The normal distribution for a single variable. The probability density function of a nor-
mally distributed random variable (the normal p.d.f.) is

fY (y) = 1 expc- 1y - m b 2 d , (17.36)
s 22p 2a s

where exp(x) is the exponential function of x. The factor 1>(s22p) in Equation (17.36)

ensures that Pr( - ∞ … Y … ∞) = ∞ fY(y)dy = 1.

1-∞
The mean of the normal distribution is m, and its variance is s2. The normal distribu-

tion is symmetric, so all odd central moments of order three and greater are zero. The

fourth central moment is 3s4. In general, if Y is distributed N(m, s2), then its even central

moments are given by