In many cases, you can model a binary communication system as follows. A 0 or 1

Question

In many cases, you can model a binary communication system as follows. A 0 or 1 is chosen as the message and the transmitter sends a signal over a medium to indicate that choice. The receiver receives a corrupted version of that signal and makes the best decision it can about whether a 0 or 1 was intended. Suppose that the message is a 1 with probability p. Also suppose that no matter what the transmitted message was, the receiver makes a random decision error about the transmitted message with a probability . Let A be the event that the intended message was a 1, and B be the event that the receiver decision was a 1.

(a) There are four possible outcomes for this experiment. List them along with their associated probabilities.

(b) Find a value for for which the intended message and received message are independent. Comment on the value of this as a practical communication system.

Explanation / Answer

Solution

Given P(Message is 1) = p, P(Message is 0) = 1 – p

Let P(Receiver makes a decision error) = q

=> P(Receiver decodes the message correctly) = 1 – q

Part (a)

The 4 possibilities and the corresponding probabilities are given below:

1. AB = Message is 1 and receiver reads it as 1 = p(1 – q)

2. ABc = Message is 1 and receiver reads it as 0 = pq

3. AcB = Message is 0 and receiver reads it as 1 = (1 – p)q

4. AcBc = Message is 0 and receiver reads it as 0 = (1 – p)(1 – q)

Part (b)

When p = q = ½, all the above probabilities are the same

=> intended message and received message are independent.

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