Problem with discrete and continuous random variables

In summary, the question is asking for the probabilities of X = 0 and X = 1, given that the output of the channel is 0.2. However, since the noise component N is a continuous distribution, it is impossible for it to take on an exact value of 0.2. Therefore, the probabilities of X = 0 and X = 1 cannot be determined with certainty, making the question ambiguous.
  • #1
lolproe
3
0

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 [itex]\mathcal{N}(0,0.1)[/itex].

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

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

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.
 
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  • #2
lolproe said:

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 [itex]\mathcal{N}(0,0.1)[/itex].

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

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

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?
 

Related to Problem with discrete and continuous random variables

1. What is the difference between discrete and continuous random variables?

Discrete random variables are those that can only take on a finite or countably infinite number of values, while continuous random variables can take on any value within a certain range. For example, the number of children in a family is a discrete random variable, while the height of a person is a continuous random variable.

2. How are probabilities calculated for discrete and continuous random variables?

For discrete random variables, probabilities are calculated by summing up the probabilities of each possible outcome. For continuous random variables, probabilities are calculated by finding the area under the probability density function within a certain range.

3. What are some examples of discrete and continuous random variables?

Examples of discrete random variables include the number of heads obtained when flipping a coin, the number of cars in a parking lot, and the number of students in a classroom. Examples of continuous random variables include the temperature of a room, the weight of a person, and the time it takes to complete a task.

4. How are discrete and continuous random variables used in real life?

Discrete and continuous random variables are used in various fields, such as finance, engineering, and healthcare, to model and analyze different types of data. For example, they can be used to predict stock prices, optimize manufacturing processes, and study the effects of a new medication.

5. What are the limitations of using discrete and continuous random variables?

One limitation is that they assume that the data follows a certain probability distribution, which may not always be the case in real life. Additionally, they may not be able to accurately capture complex or rare events. It is important to carefully consider the assumptions and limitations when using these variables in analysis.

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