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# CauchyDistribution

 CauchyDistribution represents a Cauchy distribution with location parameter a and scale parameter b.
• The probability density for value in a Cauchy distribution is proportional to . »
Probability density function:
Cumulative distribution function:
Mean and variance of a Cauchy distribution are indeterminate:
Median:
Probability density function:
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Cumulative distribution function:
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Mean and variance of a Cauchy distribution are indeterminate:
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Median:
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 Scope   (5)
Generate a set of pseudorandom numbers that are Cauchy distributed:
Compare the histogram to the PDF:
Distribution parameters estimation:
Estimate the distribution parameters from sample data:
Compare a density histogram of the sample with the PDF of the estimated distribution:
Higher-order moments are indeterminate:
Hazard function:
Quantile function:
 Applications   (1)
A pendulum hangs at height above the origin. The angle that it makes with the vertical axis is uniformly distributed from to . Find the distribution of the horizontal distance between the pendulum and the vertical axis []:
And this is CauchyDistribution, as shown:
Find the probability that the distance between the pendulum and vertical axis is at least :
Assume and find the area under the PDF plot equal to this probability:
Parameter influence on the CDF for each :
Cauchy distribution is closed under translation and scaling by a positive factor:
Cauchy distribution is closed under certain transformations:
Sum of Cauchy-distributed variates follows Cauchy distribution:
Proof based on characteristic functions:
The inverse of a Cauchy distribution centered at 0 is also a Cauchy distribution:
Relationships to other distributions:
The ratio of two normally distributed variables is a CauchyDistribution:
If is uniformly distributed, then has a CauchyDistribution:
Cauchy distribution is a limiting case of a PearsonDistribution of type 4:
CauchyDistribution is a special case of a PearsonDistribution of type 7:
Cauchy distribution is a StableDistribution:
CauchyDistribution is a singular limit of a HyperbolicDistribution of , given and :
CauchyDistribution is not defined when a is nonreal:
CauchyDistribution is not defined when b is non-positive:
Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:
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