B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

8 .2 Random Variables 41 1
Similarly,
Px(Xi) = L Px y
(Xi, YJ )
j
(8.23b)
The probabilities P x (x;) and P y
(y J
) are called marginal probabilities. Equations (8.23) show
how to determine marginal probabilities from joint probabilities. Results ofEqs. (8.20) through
(8.23) can be extended to more than two RVs.
Example 8. 1 3 A binary symmetric channel (BSC) error probability is P e· The probability of transmitting 1
is Q, and that of transmitting O is 1 - Q (Fig. 8.7). Determine the probabilities of receiving 1
and O at the receiver.
Figure 8.7
Binary symmetric
channel (BSC).
1 - P e
x = l --------➔ y = l
(Q)
x = I __ ___ _ ___ ....., y = o
( 1 - Q) 1 - P e
I
u
If x and y are the transmitted digit and the received digit, respectively, then for a BSC,
P y 1x(Oll) = P y 1x(llO) = P e
P y 1x (OIO) = P y 1x(lll) = 1 - P e
Also,
and Px (O) = 1 -Q
We need to find P y
(l) and P y
(O). From the total probability theorem,
we find
Similarly,
P y
(l) = Px (O)Py1x (llO) + Px (l)P y 1x (lll)
= (1 - Q)P e + Q(l - P e )
P y
(O) = (1 - Q)(l - P e )+ QP e
I
These answers seem almost obvious from Fig. 8.7.
Note that because of channel errors, the probability of receiving a digit 1 is not the
same as that of transmitting 1. The same is true of 0.