Tangent of Uniform Distribution has Standard Cauchy Distribution
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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