Conditional density function - need help please

1. Jun 3, 2005

ionlylooklazy

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

Last edited: Jun 3, 2005
2. Jun 3, 2005

juvenal

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. Jun 3, 2005

Hurkyl

Staff Emeritus
Whats y? n + x?

4. Jun 3, 2005

ionlylooklazy

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

Last edited: Jun 3, 2005
5. Jun 3, 2005

TenaliRaman

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. Jun 3, 2005

juvenal

http://rkb.home.cern.ch/rkb/AN16pp/node38.html [Broken]

Also - the conditional pdf f(y|x) would, intuitively to me, be a Gaussian with mean (x-2).

Last edited by a moderator: May 2, 2017
7. Jun 7, 2005

ionlylooklazy

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?

Last edited: Jun 7, 2005
8. Jun 7, 2005

juvenal

Not sure what you mean by impossible.