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Cauchy distribution

Cauchy Distribution

Probability Density FunctionThe general formula for the probability density function of the Cauchy distribution isf(x) = 1/[s*PI*(1 + ((x-t)/s)**2)]
where t is the location parameter and s is the scale parameter. The case where t = 0 and s = 1 is called the standard Cauchy distribution. The equation for the standard Cauchy distribution reduces to
f(x) = 1/(PI*(1+x**2))
Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function.
The following is the plot of the standard Cauchy probability density function.
plot of the Cauchy probability density function
Cumulative Distribution FunctionThe formula for the cumulative distribution function for the Cauchy distribution isF(x) = 0.5 + ARCTAN(x)/PI
The following is the plot of the Cauchy cumulative distribution function.
plot of the Cauchy cumulative distribution function
Percent Point FunctionThe formula for the percent point function of the Cauchy distribution isG(p) = -COT(PI*p)
The following is the plot of the Cauchy percent point function.
plot of the Cauchy percent point function
Hazard FunctionThe Cauchy hazard function can be computed from the Cauchy probability density and cumulative distribution functions.The following is the plot of the Cauchy hazard function.
plot of the Cauchy hazard function
Cumulative Hazard FunctionThe Cauchy cumulative hazard function can be computed from the Cauchy cumulative distribution function.The following is the plot of the Cauchy cumulative hazard function.
plot of the Cauchy cumulative hazard function
Survival FunctionThe Cauchy survival function can be computed from the Cauchy cumulative distribution function.The following is the plot of the Cauchy survival function.
plot of the Cauchy survival function
Inverse Survival FunctionThe Cauchy inverse survival function can be computed from the Cauchy percent point function.The following is the plot of the Cauchy inverse survival function.
plot of the Cauchy inverse survival function
Common Statistics
MeanThe mean is undefined.
MedianThe location parameter t.
ModeThe location parameter t.
RangeInfinity in both directions.
Standard DeviationThe standard deviation is undefined.
Coefficient of VariationThe coefficient of variation is undefined.
SkewnessThe skewness is undefined.
KurtosisThe kurtosis is undefined.
Parameter EstimationThe likelihood functions for the Cauchy maximum likelihood estimates are given in chapter 16 of Johnson, Kotz, and Balakrishnan. These equations typically must be solved numerically on a computer.
CommentsThe Cauchy distribution is important as an example of a pathological case. Cauchy distributions look similar to a normal distribution. However, they have much heavier tails. When studying hypothesis tests that assume normality, seeing how the tests perform on data from a Cauchy distribution is a good indicator of how sensitive the tests are to heavy-tail departures from normality. Likewise, it is a good check for robust techniques that are designed to work well under a wide variety of distributional assumptions.The mean and standard deviation of the Cauchy distribution are undefined. The practical meaning of this is that collecting 1,000 data points gives no more accurate an estimate of the mean and standard deviation than does a single point.
SoftwareMany general purpose statistical software programs support at least some of the probability functions for the Cauchy distribution.

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