X2 + X3 / X2 - X3

Question

A binary communication system is used to send one of two messages:

(i) message A is sent with probability 2/3, and consists of an infinite sequence of zeroes,

(ii) message B is sent with probability 1/3, and consists of an infinite sequence of ones.

The ith received bit is “correct" (i.e., the same as the transmitted bit) with probability 3/4, and is “incorrect" (i.e., a transmitted 0 is received as a 1, and vice versa), with probability 1/4. We assume that conditioned on any specific message sent, the received bits, denoted by Y1,Y2,… are independent.

Is Y2+Y3 independent of Y1? Is Y2-Y3 independent of Y1?

My questions:

1) Am I right that Y2 is dependent on Y1 and Y3 is dependent on Y2 and thus, on Y1?

2) How can I compute a PDF of Z1 = Y2+Y3 and Z2 = Y2-Y3?

My idea was to compute first P(Y1=0) and P(Y2=1) Then compute P(Y2=0|Y1=0), P(Y2=0|Y1=1),P(Y2=1|Y1=0), P(Y2=1|Y1=1) Then P(Y3=0|Y1=0, Y2=0)... P(Y3=1|Y1=1,Y2=1)

But even when I do it, how the joint distribution tables will differ?

I'm at a loss because I've read a lot about how to compute a PDF for two independent variables and I do know how to check if two variables are independent given their joint distribution table, but honestly for some reason this task makes me feel I'm a second-grader.

Answer 1

1. Y1, Y2, Y3 are only independent given either A or B was sent. That is P(Y1, Y2, Y3|A) = P(Y1|A) * P(Y2|A) * P(Y3|A), or equivalent given B. Intuitively, Y1, Y2, and Y3 should be dependent over all.

2. Y2+Y3 is a random variable whose values depend on the values of Y2 and Y3. So P(Y2+Y3) = P(Y2, Y3) for all combination of Y2, Y3 values. Example, P(Y2+Y3 = 0) = P(Y2=0, Y3=0). But Y2+Y3 can be {0, 1, 2}, and in order to compute the correct probability, you have to again split P(Y2=0, Y3=0) into P(Y2=0, Y3=0|A) + P(Y2=0, Y3=0|B).

To test independence of two random variables, if not able to firgure out intuitively, one can always compute the joint probability table. So instead of computing P(Y2+Y3) and P(Y1) separately, you should compute P(Y1, Y2+Y3) under all scenarios, that is given A or B was sent. Example, P(Y1=0, Y2+Y3=0) = P(Y1=0, Y2+Y3=0|A)*P(A) + P(Y1=0, Y2+Y3=0|B)*P(B)

After you fill out the joint probability table, officially, you should check if P(A,B) = P(A) * P(B), but you can simply just check the proportions given one value against another.