Conditional density function - please

In summary: The CDF should give you a value between 0 and 1 for any given input x and output y. It could be that the values you chose for y do not correspond to valid inputs for the given distributions. In summary, the conversation discusses finding an expression for the conditional density function (CDF) of a signal given the observation of the output. The signal is a random variable with an exponential distribution and a mean of 3, while the noise is a Gaussian distribution with a mean of -2 and a variance of 3. The CDF can be calculated using Bayes' theorem and the convolution of the signal and noise distributions. Simplifying the resulting expression may lead to a Gaussian distribution with a mean of (x-2
  • #1
ionlylooklazy
30
0
conditional density function - need help please!

given

a signal x, is a random variable which is expontential with a mean of 3. it is transmitted through an additive gaussian noise channel, where the gaussian noise has a mean of -2 and a variance of 3. the signal and noise are independent.


Find an expression for the CDF (conditional density function) of the signal given the observation of the output. fx(x|y)

what i think...

from bayes theorem i know:

fx(x|y) = fx(y|x)*fx(x) / fy(y)

assuming:
output = y
noise = n
input = x

y = n+x

how do i find fx(y|x) ?

the only info i have are the probability density function's for x and n

also every attempt at convoluting the exponential with the guassian (to find y) has failed whether by hand, calculator, or matlab
 
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  • #2
Since I'm headed to bed right now - I don't have time to think about this more thoroughly, but maybe you could work with the Fourier transforms and take advantage of the convolution theorem?
 
  • #3
Whats y? n + x?
 
  • #4
i assume y is is n + x, that is all the information given (the top paragraph) and the question to find the expression for CDF fx(x|y), so y should be the convolution of n and x
 
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  • #5
y is the output however you are not interpreting the signals i think?

The given signal is 3e^(-3x) and the noise is Gaussian(-2,3)
The output is additive which means,
y = 3e^(-3x) + Gaussian(-2,3)
Now can u find f(y|x) ?

-- AI
 
  • #6
Some follow ups:
The convolution of an exponential and a and a normal distribution is approximated by another exponential distribution.

http://rkb.home.cern.ch/rkb/AN16pp/node38.html

Also - the conditional pdf f(y|x) would, intuitively to me, be a Gaussian with mean (x-2).
 
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  • #7
ah of course, many thanks


also, after i found everything i simplified fx(x|y) and plotted it for the cases y = {-5 -1 0 1 5 10} but only the cases y = {0 1} turned out something resembling a probability density function, would this just be that it is impossible to determine x for these cases?
 
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  • #8
Not sure what you mean by impossible.
 

1. What is a conditional density function?

A conditional density function is a statistical concept that describes the probability distribution of a random variable given that another random variable has a specific value. It is used to model the relationship between two variables and can be used to make predictions and in statistical inference.

2. How is a conditional density function different from a marginal density function?

A marginal density function describes the probability distribution of a single random variable, while a conditional density function describes the probability distribution of a random variable given that another random variable has a specific value. In other words, a marginal density function considers all possible values of a variable, while a conditional density function only considers specific values based on the value of another variable.

3. What is the formula for a conditional density function?

The formula for a conditional density function is f(x|y) = f(x,y)/f(y), where f(x,y) is the joint density function and f(y) is the marginal density function of y. In simpler terms, the conditional density function is equal to the probability of x and y occurring together, divided by the probability of y occurring on its own.

4. How is a conditional density function used in statistics?

Conditional density functions are used in statistics to model the relationship between two variables and to make predictions about the value of one variable based on the value of another. They are also used in statistical inference to test hypotheses and make inferences about populations based on sample data.

5. What are some common applications of conditional density functions?

Conditional density functions are commonly used in fields such as economics, finance, and engineering to model relationships between variables and make predictions. They are also used in machine learning and data analysis to understand and make predictions about complex datasets. Additionally, they are used in statistical process control to monitor and improve processes in manufacturing and other industries.

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