Page 109 - Fundamentals of Communications Systems

Review of Probability and Random Variables 3.23

Equation (3.30) can be simplified using the following two well-known linear
algebra identities:

1.

AB −1
det = det[A − BD C]det[D] (3.31)
CD
2.

−1 −1
−1
AB A 0 −A B −1 −1 −1
= + [D − CA B] −CA I
CD 00 I
00 I −1 −1 −1

= −1 + −1 A − BD C] [ I −BD (3.32)
0 D −D C
where A, B, C, and D are arbitrary matrices. Using Eq. (3.31) the ratio of the
determinants in Eq. (3.30) is given as

det[C Z ] −1
= det C X − C XY C C YX = det[C X|Y ] (3.33)
Y
det[C Y ]
Using Eq. (3.32) to reformulate C Z in Eq. (3.30) gives
1 T −1 1 T −1

y
exp − [ z − m Z ] C [ z − m Z ] = exp − [ − m Y ] C [ y − m Y ]
Y
Z
2 2

1 T −1
× exp − F G F (3.34)
2
where
−1
F = − m X − C XY C [ y − m Y ] = x − E[X|Y = y ]
x

Y
and
−1
G = C X − C XY C C T XY = C X|Y
Y
Using Eq. (3.33) and Eq. (3.34) in Eq. (3.30) and cancelling the common terms
gives the desired result.
3.4 Homework Problems
Problem 3.1. This problem exercises axiomatic definitions of probability. A com-
munications system used in a home health care system needs to communicate
four patient services, two digital and two analog. The two digital services are
a 911 request and a doctor appointment request. The two analog services are
the transmission of an electrocardiogram (EKG) and the transmission of audio
output from a stethoscope. The patient chooses each of the services randomly
depending on the doctor’s prior advice and the patient’s current symptoms.
Assume only one service can be requested at a time.