My Math Forum moments of Cauchy distribution

 February 21st, 2011, 07:45 AM #1 Member   Joined: Dec 2010 From: Drachten, the Netherlands Posts: 49 Thanks: 0 moments of Cauchy distribution show that for a Cauchy distribution the first moment and second moment about the origin does not exist. a Cauchy distribution has the following distribution: f(x) = (B/pi)/((x-a)^2+b^2) for -infinity
 February 21st, 2011, 05:15 PM #2 Senior Member   Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3 Re: moments of Cauchy distribution Setting a=0 and ignoring the 1/pi factor: $\int_{-\infty}^\infty \frac{bx}{x^2+b^2}\mathrm{d}x=\int_{-\infty}^0 \frac{bx}{x^2+b^2}\mathrm{d}x+\int_0^\infty \frac{bx}{x^2+b^2}\mathrm{d}x$ Let u = x²+b², du = 2x dx $\int_0^\infty \frac{bx}{x^2+b^2}\mathrm{d}x=\frac{b}{2}\int_{b^2 }^\infty \frac{\mathrm{d}u}{u} = \frac{b}{2}\lim_{r\to\infty}\log r - 2\log b$ which doesn't exist. Break the integrals in half and use substitutions to simplify them.

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# moment for Cauchy distribution

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