Quotient of Gaussian Distributions has Cauchy Distribution/Corollary

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