Conditional density function - please

[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

 

[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?

 

[Hurkyl]

Whats y? n + x?

 

[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

 

[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

 

[juvenal]

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).

 

[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?

 

[juvenal]

Not sure what you mean by impossible.