Tangent of Uniform Distribution has Standard Cauchy Distribution

Theorem

Let $X$ be a continuous random variable with a uniform distribution on the closed real interval $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$:

$X \sim \ContinuousUniform {-\dfrac \pi 2} {\dfrac \pi 2}$

Let $Y$ be a continuous random variable such that:

$Y = \tan X$

where $\tan$ denotes the tangent function.

Then $Y$ has the standard Cauchy distribution:

$Y \sim \Cauchy 0 1$

Proof

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Examples

Radiation Particles

Let particles be emitted from a source of radiation $S$ in a plane in random (equally likely) directions.

Let these particles travel in a straight line to a plane collector some distance from $A$.

Then the distribution of the points of impact of the particles on the collector has a standard Cauchy distribution.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Cauchy distribution