Question

The input X to a binary communication channel assumes the values +1
or −1 with probabilities 1/3 and 2/3 respectively. The output Y of the AWGN channel is given by
Y = X + N where N is zero mean Gaussian noise with variance 2
n = 1.
1. Find the conditional pdf of Y given X = +1.

Answer 1

@phi

Answer 2

so obiviously
Y=X+N
X being 1

Y=1+N

Y=1+N

N being \[\phi(x-\mu/1)\]
\[\phi(x-0/1)\]

\[\phi(1-0/1)\]=\[\phi(1)\]

we add to 1 on top so

\[\phi(1)+1\]

and divde by conditional probability
1/3

Answer 3

I have not looked at this stuff, but my intuition is that if x is given as 1, the pdf for y should look like N, but centered at 1 (a gaussian with 1 mean, variance 2)

Answer 4

I'll put this subject on my "to do" list. In the meantime....

Answer 5

I think @phi is correct. As long as X is given as 1, then the probability of x occurring doesn't affect the pdf.

Answer 6

A more interesting (read: "hard") pdf is the overall pdf that encompasses both x = 1 and x = -1. Thankfully, that wasn't the question :)

Answer 7

my "to do" list seems to grow with every day I sign on here... (sigh)

Answer 8

"then the probability of x occurring doesn't affect the pdf."

can you explain why?

Answer 9

@satellite73

Answer 10

I thought of it just like simple conditional probability... if P(x) = 0.5 and P(y) = 0.3, but the problem says, "Given x, what is the probability of y?", then the P(x) doesn't affect the probability of P(x|y), since you know x is given.

Here, the fact that x = 1 occurs 1/3 of the time is irrelevant to the pdf if you are given that x = 1.

Answer 11

https://onlinecourses.science.psu.edu/stat414/book/export/html/95 This site might help you with Conditional Distributions. It might help you.

Answer 12

thank you so much for not ignoring my question

Answer 13

Your welcome. anytime.

Answer 14

thanks to jessica as well

Answer 15

So, I think if you run "x = 1" through the channel, then you would expect Y to look like 1, but it would have the shape of the pdf of the AWGN due to the channel. But the signal X = 1 would move the mean to be around 1 vs. around 0 which is what the channel looks like with no signal.

Answer 16

My comment earlier about the overall pdf was if you were trying to build a sensor/detector, and you didn't know whether X was +1 or -1, and you picked up an output signal Y = 0.2, you would have to decide if it was really an X= -1 input or if it was an X = + 1 input... the conditional pdfs would be overlapping.

Answer 17

ok,