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Math Help - Pareto PDF

  1. #1
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    Pareto PDF

    Can someone show me how this integrates to 1?

    \int_{-\infty}^{\infty} \frac{\gamma \theta^{\gamma}}{x^{\gamma +1}}

    Also, can you show me how to derive the variance and quartiles? Thank you.
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  2. #2
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    Quote Originally Posted by VENI View Post
    Can someone show me how this integrates to 1?

    \int_{-\infty}^{\infty} \frac{\gamma \theta^{\gamma}}{x^{\gamma +1}}

    Also, can you show me how to derive the variance and quartiles? Thank you.
    Since this is a Pareto distribution:

    1. The support is [\theta, + \infty). So you need to show that \int_{\theta}^{+ \infty} \frac{\gamma \theta^{\gamma}}{x^{\gamma +1}} \, {\color{red} dx} = 1.

    (*Ahem .... note what is in red by the way).

    2. You should realise that \gamma > 0 and \theta > 0. Knowing \gamma > 0 is of particular importance in solving the integral.


    Now you should note that \int_{\theta}^{+ \infty} \frac{\gamma \theta^{\gamma}}{x^{\gamma +1}} \, {\color{red} dx} = \gamma \theta^{\gamma} \int_{\theta}^{+ \infty} \frac{1}{x^{\gamma +1}} \, {\color{red} dx} = \gamma \theta^{\gamma} \int_{\theta}^{+ \infty} x^{-(\gamma +1)} \, {\color{red} dx}.

    So you have a simple improper integral to calculate.

    --------------------------------------------------------------------------------------------------------------------------

    As for the variance, you should know that Var(X) = E(X^2) - [E(X)]^2. So set up the necessary integrals and calculate the required expectations. Again, you have simple improper integrals.

    --------------------------------------------------------------------------------------------------------------------------

    As for the quartiles:

    To find the median Q_2 you need to solve \gamma \theta^{\gamma} \int_{\theta}^{Q_2} x^{-(\gamma +1)} \, {\color{red} dx} = \frac{1}{2} for Q_2. The integral is simple to do and the subsequent equation is simple to solve.

    The other quartiles are found in a similar way.
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