The ratio of independent normally distributed variates with zero mean is distributed with a Cauchy distribution. This can be seen as follows. Let and both have mean 0 and standard deviations of and , respectively, then the joint probability density function is the bivariate normal distribution with ,
(1)
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From ratio distribution, the distribution of is
(2)
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(3)
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(4)
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But
(5)
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so
(6)
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(7)
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(8)
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which is a Cauchy distribution.
A more direct derivative proceeds from integration of
(9)
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(10)
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where is a delta function.