this is more of a theoretical question as the implementation doesn't really matter.

I'm experimenting in Matlab, and I was curious about something. I know that the division of Gaussian-distributed random variates (with mean = 0) results in a Cauchy distribution.

(See here: Normal Ratio Distribution on Wolfram Mathworld)

But if I take the Fourier Transform of each Gaussian-distribution from which I'm getting my variates, is that equal to the Fourier Transform of a Cauchy distribution?

i.e. Is fft(G1)/fft(G2) = fft(equivalent Cauchy)

If not, how do I model the ratio of two normally distributed variates in the frequency domain?


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    $\begingroup$ I think this is actually a question for statistics.se $\endgroup$ – Jonas Jan 21 '16 at 17:18
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    $\begingroup$ Consider the case where both Gaussians have the same distribution, whence their FTs must be the same. Consequently their ratio will be $1$ (at least wherever they are nonzero--which is everywhere on the Real line). Is that the FT of a Cauchy distribution? What is it the FT of? What does that tell you about your conjecture? Finally, since you know the ratio is a (scaled) Cauchy, why not just look up or directly calculate its FT? $\endgroup$ – whuber Jan 21 '16 at 20:07
  • $\begingroup$ The ratio of Fourier transforms of the densities (or equivalently, the characteristic functions) of two (independent) random variables $X$ and $Y$ is hardly ever the Fourier transform of the density of $\frac XY$. In fact, the closest result is that the Fourier transform of the density of the sum $X+Y$ is the product of the Fourier transforms of the densities of $X$ and $Y$. In other words, your hypothesized relationship is not true, and there is no simple general formula of the kind that you seek. $\endgroup$ – Dilip Sarwate Jan 21 '16 at 23:45

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