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Math Help - Truncated prior distribution

  1. #1
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    Truncated prior distribution

    Can anyone show me how to set out this problem so i can tackle it myself?

    Suppose that a random sample of size n = 100 from an exponential distribution with parameter \theta is about to be collected. It is thought that a Ga(2, 2) prior distribution for \theta is appropriate but, since it is believed that values of \theta exceeding 6 are impossible, the prior distribution is truncated above 6.

    Calculate the prior probability that \theta exceeds 6 assuming the truncated prior distribution. What is the probability when the prior distribution is not truncated?

    Thanks!
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  2. #2
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    I think what they intend for you to do is renormalize the gamma so that it has support (0, 6). So, you would want to find a k so that

    \displaystyle<br />
\int_0 ^ 6 k x e^{x/2} \ dx = 1<br />

    and then take your prior to be \pi(\theta) = k \theta e^{\theta / 2} I(0 < \theta < 6).

    (Note: there are different parameterizations of the gamma. I am using the one where Gamma(a, b) implies that the mean of the distribution is ab.)
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    Ok so then i can do that integration and i find that k = \frac{1}{8e^{3}+4}. So is I the fisher matrix?
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    Also, i think actually i am more used to seeing the parameterization of the gamma where the mean is a/b.

    So assumably this would change my integral?
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  5. #5
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    I is the indicator function. It is equal to 1 when its argument is true, and 0 otherwise.

    And yes, the integral would change if you are using a different parameterization. I hate the one where the mean is a/b.
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    Ok so what do i need to do to form my integral in order to find k?
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  7. #7
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    Just integrate the gamma density from 0 to 6 and normalize it so that it integrates to 1.
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    Ok sorry so i dont know which gamma density i am integrating....? because its got to be different to the 1 you suggested i guess? What was the difference between "the gamma density" and the one you suggested?
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    ok iv managed to find what k is and so i beleive iv got my prior distribution..... but i dont understand the question asks "Calculate the prior probability that \theta exceeds 6 assuming the truncated prior distribution". This confuses me, as isnt the probability 0 because we have truncated it. i.e. i have a pdf for 0 < \theta< 6 and then i have 0, otherwise? Help!
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  10. #10
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    Quote Originally Posted by sirellwood View Post
    ok iv managed to find what k is and so i beleive iv got my prior distribution..... but i dont understand the question asks "Calculate the prior probability that \theta exceeds 6 assuming the truncated prior distribution". This confuses me, as isnt the probability 0 because we have truncated it. i.e. i have a pdf for 0 < \theta< 6 and then i have 0, otherwise? Help!
    Yes, that is odd. They might have meant to say the probability that X exceeds 6.
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