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Math Help - conditional normal distribution

  1. #1
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    conditional normal distribution

    Assume two random variables X and Y are not independent,

    if P(X), P(Y) and P(Y|X) are all normal, then does P(X|Y) also can only be normal or not necessarily?

    thanks.
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  2. #2
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    Re: conditional normal distribution

    Hey simon999.

    Did you ask this question on physics forums? If so I gave an answer there.
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  3. #3
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    Re: conditional normal distribution

    Hi yes I did, not sure I understood your replies though sorry,

    so is your answer that P(X|Y) does also has to be normal or not necessarily?
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  4. #4
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    Re: conditional normal distribution

    No it has to be if P(Y|X) is proportional to P(X|Y)*P(Y) and if P(X|Y) and P(Y) are both Normal then P(X|Y) also has to be normal.

    For a more indepth explanation look up Bayesian statistics with Normal Likelihood and a Normal conjugate prior to see the details.
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  5. #5
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    Re: conditional normal distribution

    Are you sure that by having both marginals normal already cover your condition of P(Y|X) is proportional to P(X|Y)*P(Y)
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  6. #6
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    Re: conditional normal distribution

    If you want to prove it, take two PDFs that are Normally distributed f(X) and g(X) and then show that f(X)*g(X) also has a normal distribution. This is a very simple exercise and would do you well to understand what is going on.
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  7. #7
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    Re: conditional normal distribution

    Thanks,

    Someone else tells me


    Look at the 2-d density function for dependent normal variables. You will see that, except for singular cases, that the form of the integrand is symmetric in x and y (except for constants). Therefore if one of the conditional distributions is normal, the other must be.

    Singular case - correlation = 1.

    Is he wrong then?
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  8. #8
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    Re: conditional normal distribution

    You can have a variable covariance matrix in the general situation (can you can include this in your PDF's for X and Y if you want).
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  9. #9
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    Re: conditional normal distribution

    Sorry so what is the right answer?

    if P(X), P(Y) and P(Y|X) are all normal, then does P(X|Y) also can only be normal or not necessarily?

    You are saying no but the other person is saying yes
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