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February 21st, 2011, 07:45 AM   #1
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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<x< infinity

I tried to find
E(X)= integral from -infinity to infinity of x(B/pi)/((x-a)^2+b^2)
and E(X^2)= integral from -infinity to infinity of x^2(B/pi)/((x-a)^2+b^2)

However the integral is to hard for me to solve.
Can someone help me with this or is there some other method to prove that these moment does not exists?

Thanks in advance
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February 21st, 2011, 05:15 PM   #2
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Re: moments of Cauchy distribution

Setting a=0 and ignoring the 1/pi factor:


Let u = x+b, du = 2x dx
which doesn't exist.
Break the integrals in half and use substitutions to simplify them.
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