3.5. The Multivariate Normal Distribution 207What about the other components,Y 2 ,...,Yn? As the following theorem shows,
they share a similar property relative to the order of their associated eigenvalue.
For this reason, they are called thesecond,third, through thenth principal
components, respectively.
Theorem 3.5.5. Consider the situation described above. Forj=2,...,nand
i=1, 2 ,...,j− 1 ,Var[a′X]≤λj=Var(Yj), for all vectorsasuch thata⊥viand
‖a‖=1.
The proof of this theorem is similar to that for the first principal component
and is left as Exercise 3.5.20. A second application concerning linear regression is
offered in Exercise 3.5.22.EXERCISES3.5.1.LetXandYhave a bivariate normal distribution with respective parameters
μx=2. 8 ,μy= 110,σ^2 x=0. 16 ,σy^2 = 100, andρ=0.6. Using R, compute:(a)P(106<Y <124).(b)P(106<Y < 124 |X=3.2).3.5.2.LetX andY have a bivariate normal distribution with parametersμ 1 =
3 ,μ 2 =1,σ^21 =16,σ 22 = 25, andρ=^35. Using R, determine the following
probabilities:
(a)P(3<Y <8).(b)P(3<Y < 8 |X=7).(c)P(− 3 <X<3).(d)P(− 3 <X< 3 |Y=−4).3.5.3.Show that expression (3.5.4) is true.
3.5.4.Letf(x, y) be the bivariate normal pdf in expression (3.5.1).
(a)Show thatf(x, y) has an unique maximum at (μ 1 ,μ 2 ).(b)For a givenc>0, show that the points{(x, y):f(x, y)=c}of equal proba-
bility form an ellipse.3.5.5.LetXbeN 2 (μ,Σ). Recall expression (3.5.17) which gives the probability of
an elliptical contour region forX. The R function^8 ellipmakeplots the elliptical
contour regions. To graph the elliptical 95% contour for a multivariate normal
distribution withμ=(5,2)′andΣwith variances 1 and covariance 0.75, use the
code
(^8) Part of this code was obtained from an annonymous author at the site
http://stats.stackexchange.com/questions/9898/