Problem with discrete and continuous random variables

[lolproe]

Homework Statement

A binary information source produces 0 and 1 with equal probability. The output of the source, denoted as X, is transmitted via an additive white Gaussian noise (AWGN) channel. The output of the channel, denoted as Y, satisfies Y = X + N, where the random noise N has the distribution \mathcal{N}(0,0.1).

a) If the channel outputs 0.2, what is the probability that 0 is the source output and what is the probability that 1 is the source output

b) If the channel output is 0.2 and you are required to make a guess on the source output, what is your guess and why?

Homework Equations

Not entirely sure

The Attempt at a Solution

I've tried framing this question a few different ways, but every time I can't make sense of it.

Setting it up exactly like the question is asked, it's simply asking to find the probabilities that X = 0 and X = 1, given that Y = 0.2. To solve that, I think I would need to find the conditional distribution of X, using

f_{X|Y}(x|y) = \frac{f(x,y)}{f_Y(y)}

But for that, I need the joint PDF of X and Y. As far as I can tell, X would be a Bernoulli distribution, but since N is a normal distribution, I don't know how you can find their sum in the first place to get a PDF for Y at all. I've never seen an example of adding a discrete and continuous RV together and I don't really understand how it would work at all, so this is where this approach falls apart for me.

The more I look at the question though, the more I think there is no answer though. Instead of solving it like above, I tried looking at it more generally. The source will only produce an output of exactly 1 or exactly 0, and the output from the channel is exactly 0.2. For this to be satisfied, wouldn't the noise component need to be exactly 0.2 or 0.8? And since the noise is a continuous distribution, wouldn't it have 0 probability (by definition) of taking on an exact value? The fact that part b) implies that you might get an ambiguous answer from part a) makes me think this might be right, but I'm not really confident in my logic arriving at this step. It is obviously more likely for the noise to equal 0.2 instead of 0.8, but I just don't know if it's possible to figure out the relationship exactly.

 

[Ray Vickson]

Homework Statement

A binary information source produces 0 and 1 with equal probability. The output of the source, denoted as X, is transmitted via an additive white Gaussian noise (AWGN) channel. The output of the channel, denoted as Y, satisfies Y = X + N, where the random noise N has the distribution \mathcal{N}(0,0.1).

a) If the channel outputs 0.2, what is the probability that 0 is the source output and what is the probability that 1 is the source output

b) If the channel output is 0.2 and you are required to make a guess on the source output, what is your guess and why?

Homework Equations

Not entirely sure

The Attempt at a Solution

I've tried framing this question a few different ways, but every time I can't make sense of it.

Setting it up exactly like the question is asked, it's simply asking to find the probabilities that X = 0 and X = 1, given that Y = 0.2. To solve that, I think I would need to find the conditional distribution of X, using

f_{X|Y}(x|y) = \frac{f(x,y)}{f_Y(y)}

But for that, I need the joint PDF of X and Y. As far as I can tell, X would be a Bernoulli distribution, but since N is a normal distribution, I don't know how you can find their sum in the first place to get a PDF for Y at all. I've never seen an example of adding a discrete and continuous RV together and I don't really understand how it would work at all, so this is where this approach falls apart for me.

The more I look at the question though, the more I think there is no answer though. Instead of solving it like above, I tried looking at it more generally. The source will only produce an output of exactly 1 or exactly 0, and the output from the channel is exactly 0.2. For this to be satisfied, wouldn't the noise component need to be exactly 0.2 or 0.8? And since the noise is a continuous distribution, wouldn't it have 0 probability (by definition) of taking on an exact value? The fact that part b) implies that you might get an ambiguous answer from part a) makes me think this might be right, but I'm not really confident in my logic arriving at this step. It is obviously more likely for the noise to equal 0.2 instead of 0.8, but I just don't know if it's possible to figure out the relationship exactly.

Since N is "continuous", the probability of having output = 0.2 exactly is zero. Nevertheless, 0.2 was observed!

To make sense of this, say that the observed output is in an interval (0.2-h, 0.2+h) where h > 0 is very small. That interval has nonzero probability. Given that you observe it, what are the chances that X = 0 or X = 1?