Quotient of Gaussian Distributions has Cauchy Distribution/Corollary
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Theorem
Let $X$ and $Y$ be independent continuous random variables each with a Gaussian distribution with:
- zero expectation
- the same variance $\sigma$:
\(\ds X\) | \(\sim\) | \(\ds \Gaussian 0 {\sigma^2}\) | ||||||||||||
\(\ds Y\) | \(\sim\) | \(\ds \Gaussian 0 {\sigma^2}\) |
Let $U$ be the continuous random variable defined as:
- $U = \dfrac X Y$
Then $U$ has the Cauchy distribution:
- $U \sim \Cauchy 0 1$
and so does $\dfrac 1 U$:
- $\dfrac 1 U \sim \Cauchy 0 1$
Proof
From Quotient of Gaussian Distributions has Cauchy Distribution:
- $U \sim \Cauchy 0 \lambda$
where:
- $\lambda = \dfrac {\sigma_x} {\sigma_y}$
and such that:
- $\sigma_x = \sigma_y = \sigma$
Hence $\lambda = 1$ and the result follows.
Similarly we have:
- $\dfrac 1 U = \dfrac Y X$
and again the result follows.
$\blacksquare$
Sources
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Cauchy distribution