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Math Help - Probability Problem

  1. #1
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    Probability Problem

    Hello,

    I need to prove that the next function is a density function, meaning, to prove that the integral equals 1. I know that I need to use Cauchy's distribution , but I can't figure out how exactly, you can say it's a calculus problem....

    thanks

    the function is:

    f(x)= (1/pi) * ( beta / beta^2+(x-a)^2)

    beta does not refers to the beta function, just the greek letter
    Last edited by WeeG; December 11th 2006 at 07:28 AM. Reason: Can't upload the file
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by WeeG View Post
    Hello,

    I need to prove that the next function is a density function, meaning, to prove that the integral equals 1. I know that I need to use Cauchy's distribution , but I can't figure out how exactly, you can say it's a calculus problem....

    thanks

    the function is:

    f(x)= (1/pi) * ( beta / beta^2+(x-a)^2)

    beta does not refers to the beta function, just the greek letter
    <br />
I=\int_{-\infty}^{\infty}\,f(x)dx=\frac{\beta}{\pi}\,\int_{-\infty}^{\infty}\,\frac{1}{\beta^2+(x-a)^2}dx<br />

    ...... <br />
=\frac{1}{\beta\, \pi}\,\int_{-\infty}^{\infty}\,\frac{1}{1+((x-a)/\beta)^2}dx<br />

    Now put x'=(x-a)/\beta, then:

    <br />
I=\frac{1}{\pi}\,\int_{-\infty}^{\infty}\,\frac{1}{1+x'^2}dx'<br />

    Now the integral is a standard integral and is equal to \pi,
    so:

    I=1, and so f is a density.

    RonL
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  3. #3
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    Thanks, superb solution
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