# Conditional density function - need help 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

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#### 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?

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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

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#### 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

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 [Broken]

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

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#### 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?

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#### juvenal

Not sure what you mean by impossible.

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