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748 Chapter 17 The Theory of Linear Regression with One Regressor

E(Y - m)k = k! 2)! sk (k even). (17.37)
2k>2(k >

When m = 0 and s2 = 1, the normal distribution is called the standard normal distribu-

tion. The standard normal p.d.f. is denoted f, and the standard normal c.d.f. is denoted Φ.
y22)
Thus the standard normal density is f(y) = 1 exp (- and Φ(y) = y f(s)ds.
22p
1-∞

The bivariate normal distribution. The bivariate normal p.d.f. for the two random vari-
ables X and Y is

gX,Y(x, y) 1 * exp e 1 x - mX 2 -
2psXsY 21 - r2XY) c a sX
= - rX2 Y - 2(1 b

2rXY a x - mX b a y - mY b + y - mY 2 d f , (17.38)
sX sY a sY
b

where rXY is the correlation between X and Y.
When X and Y are uncorrelated (rXY = 0), gX,Y(x, y) = fX(x)fY(y), where f is the

normal density given in Equation (17.36). This proves that if X and Y are jointly
normally distributed and are uncorrelated, then they are independently distributed.
This is a special feature of the normal distribution that is typically not true for other
distributions.

The multivariate normal distribution extends the bivariate normal distribution to
handle more than two random variables. This distribution is most conveniently stated using
matrices and is presented in Appendix 18.1.

The conditional normal distribution. Suppose that X and Y are jointly normally distrib-
uted. Then the conditional distribution of Y given X is N(mY͉X, sY2 ͉X), with mean
mY͉X = mY + (sXY>sX2 )(X - mX) and variance sY2 ͉X = (1 - r2XY)s2Y. The mean of this
conditional distribution, conditional on X = x, is a linear function of x, and the variance

does not depend on x (Exercise 17.11).

Related Distributions

The chi-squared distribution. Let Z1, Z2, c, Zn be n i.i.d. standard normal random vari-
ables. The random variable

n

W = a Zi2 (17.39)

i=1

has a chi-squared distribution with n degrees of freedom. This distribution is denoted x2n.
Because E(Z2i ) = 1 and E(Z4i ) = 3, E(W) = n and var(W) = 2n.