Page 91 - Fundamentals of Communications Systems

Review of Probability and Random Variables 3.5

event B is observed is
P [B|A j ]P [A j ] P [B|A j ]P [A j ]
P [A j |B] = = N
P [B] P [B ∩ A i ]
i=1
Proof: The definition of conditional probability gives

P [A j ∩ B] = P [A j |B]P [B] = P [B|A j ]P [A j ]

Rearrangement and total probability complete the proof.

EXAMPLE 3.7
(Binary symmetric channel) A facsimile machine divides a document up into small
regions (i.e., pixels) and decides whether each pixel is black or white. Reasonable a
priori statistics for facsimile transmission is

P [A pixel is white] = P [W] = 0.8 P [A pixel is black] = P [B] = 0.2

This pixel value is transmitted across a telephone line and the receiving fax machine
makes a decision about whether a black or white pixel was sent. Figure 3.1 is a sim-
plified representation of this operation in an extremely noisy situation. If a black pixel
is decoded (BD) what is the probability a white pixel was sent, P (W|BD)? Bayes rule
gives a straightforward solution to this problem, i.e.,

P [BD|W]P [W]
P (W|BD) = (3.12)
P [BD]
P [BD|W]P [W]
=
P [BD|W]P [W] + P [BD|B]P [B]
(0.1)0.8
= = 0.3077 (3.13)
(0.1)(0.8) + (0.9)(0.2)

[
.
White PWDW] = 09 White
Pixel Pixel
Sent Decoded
[ [
.
.
PBDW] = 01 PWDB] = 01
Black Black
Pixel Pixel
[
Sent PBDB] = 09 Decoded
.
Figure 3.1 The binary symmetric channel.