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Conditional density function - need help please

  1. Jun 3, 2005 #1
    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. jcsd
  3. Jun 3, 2005 #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?
     
  4. Jun 3, 2005 #3

    Hurkyl

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    Whats y? n + x?
     
  5. Jun 3, 2005 #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
     
    Last edited: Jun 3, 2005
  6. Jun 3, 2005 #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
     
  7. Jun 3, 2005 #6
    Some follow ups:
    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).
     
  8. Jun 7, 2005 #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?
     
    Last edited: Jun 7, 2005
  9. Jun 7, 2005 #8
    Not sure what you mean by impossible.
     
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