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Thread: Name of distribution with density proportional to incomplete gamma function

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    Name of distribution with density proportional to incomplete gamma function




    Hi,

    Does anyone know the name of the probability distribution with the following density function?

    f_X(x)=\frac{\Gamma[\alpha,x]}{\Gamma[\alpha+1]},

    where x>0, \alpha>0, \Gamma[\alpha,x] is an upper incomplete gamma function, and \Gamma[\alpha] is a gamma function.

    I occasionally found this distribution as a marginal of a pseudo Wishart distribution. I want to read papers about its property analysis, but I have trouble to find one because I don't know the name. Please help me!

    Best

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    Re: Name of distribution with density proportional to incomplete gamma function

    I have not find the name; but note that

    f_X(x) = \frac {1} {\Gamma(\alpha+1)}\int_x^{+\infty} t^{\alpha-1}e^{-t}dt

    = \frac {1} {\alpha} \int_x^{+\infty} \frac {1} {\Gamma(\alpha)}t^{\alpha-1}e^{-t}dt

    = \frac {1} {\alpha} [1 - F_Y(x)] \text{ where } Y \sim \text{Gamma}(\alpha, 1)

    = \frac {1 - F_Y(x)} {E[Y]}

    From this we see that for each positive continuous random variable Y with expectation exist, we can form another pdf by

    f_X(x) = \frac {1 - F_Y(x)} {E[Y]}

    which must integrate to 1 as

    E[Y] = \int_0^{+\infty} [1 - F_Y(x)]dx

    http://en.wikipedia.org/wiki/Expecte...ral_definition

    And from wiki, this identity is somewhat related to the layer cake representation. Not sure if this is helpful in searching the name.

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    Re: Name of distribution with density proportional to incomplete gamma function


    Hi BGM,

    Thank you for your comments. I was also aware of the similarity with the gamma cumulative density, but I enjoyed your proof a lot.

    Cheers

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