A certain binary PCM system transmits the two binary states...

A certain binary PCM system transmits the two binary states X = +1, and X = -1 with equal probability. However, because of channel noise, the receiver makes recognition errors. Also, as a result of path distortion, the receiver may lose necessary signal strength to make any decision. Thus, there are three possible receiver states: Y = +1, Y = 0, and Y = -1. Y=0 corresponds to "loss of signal". Assume that: P(Y=-1|X = +1) = 0.1, P(Y=+1|X = -1) = 0.2, and P(Y=0|X = +1) = P(Y=0|X = -1) = 0.05 (a) Find the probability P(Y=+1), P(Y=-1), and P(Y=0). (b) Find the probability P(X=+1|Y = +1) and P(X=-1|Y = -1)

Transcript

00:01 So in this question, we're given a few conditional probabilities, where y is our receiver state and x is our transmitted state.

00:12 So we want to find the probability that y is equal to plus one.

00:16 This is going to be the sum over x, probability y equals plus one given x equals x, times the probability that x is equal to x.

00:27 And these are equal probabilities.

00:29 So we're told that the probability x equals 1 is equal to the probability x equals minus 1, which is a half.

00:40 So this probability here is going to be half times the sum over the different conditional probabilities where y is equal to 1.

00:51 So we have the probability y equals 1 given x equals minus 1 times a half, plus a half times the probability y equals 1 given x equals minus 1.

01:09 So this is going to be, so we've got the probability that y equals 0 given x equals given x equals 1.

01:18 We've got the probability y is minus 1 given x equals 1.

01:22 So the probability that y is plus 1 given x equals 1 is going to be 1 minus this 0 .0 .0 .05.

01:34 And for this one we have a similar thing.

01:37 So we've got 0 .05 and 0 .2 of it being wrong.

01:42 So the probability of it, under the probability y is 1 when x is minus 1 is 0 .2.

01:48 So we got a half of 0 .2 plus a half of 0 .85.

01:57 So this is 0 .1 plus 0 .5 times 0 .85, which is 0 .525.

02:09 Oh, we need to find the probability why is minus 1 and the probability y equals 0 as well.

02:13 So this is the probability y is plus 1...

Question

Please give Ace some feedback