Binary Symmetric Channel

Consider the following simple model for a communication channel with input X and output Y.

As indicated, the transition probabilities are

$$\begin{array} {ll} P(Y=0\vert X=0) = 0.2 & P(Y=0\vert X=1) = 0.7 \\ P(Y=1\vert X=0) = 0.8 & P(Y=1\vert X=1) = 0.3. \end{array}$$

The transmission probabilities are P(X=0)=0.6 and P(X=1)=0.4, respectively.

1.

Assume that Y=0 was observed. What is the probability that X=0 was transmitted? What is the probability that X=1 was transmitted?

2.

If Y=0 wass observed which is more likely, X=0 was transmitted or X=1 was transmitted?

3.

Repeat part a) under the assumption that Y=1 was observed.

4.

Based on the observation of the output Y, a receiver for the above channel has to decide whether X=0 or X=1 was transmitted. For that purpose, it employs decision rules of the form: ``If Y=a was observed then X=b was most likely transmitted.'' Specify ``reasonable'' decision rules for the two possible values of Y.

5.

Receivers are measured by the probability of making a wrong decision. More precisely, the probability of error P_e is defined as

$$\begin{array} {cl} P_e = & P(X=0) \cdot P(\mbox{receiver dec... ...=1) \cdot P(\mbox{receiver decides $X=0$}\vert X=1).\end{array}$$

Find the probability of error for your receiver from part d).