Cauchy Probability Density

The Cauchy distribution describes ratios of normal variables and exhibits extremely heavy tails. Its probability density function is

$f\left(x\right)=\frac{1}{\pi \gamma \left(1+{\frac{x-x_0}{\gamma }}^2\right)}$

Unlike the normal distribution, the Cauchy distribution lacks finite mean or variance. That means sample averages do not converge to a typical value even for very large samples. Because the density decays only as $\mathrm{|x|}^\{-2\}$, extremely large observations remain probable. This property makes the Cauchy distribution an important counterexample in probability theory and underscores the importance of checking assumptions behind statistical estimators.

Distribution Function

The cumulative distribution function (CDF) has a closed form involving the arctangent:

$F\left(x\right)=\frac{1}{\pi }\arctan \left(\frac{x-x_0}{\gamma }\right)+\frac{1}{2}$

This step function shows that half of the probability mass lies to the left of the location parameter $x_0$. The heavy tails mean the distribution is more spread out than a normal distribution with comparable scale.

The Cauchy distribution arises in various contexts from resonance behavior in physics to Bayesian statistics. Its long tails sometimes model phenomena with rare but extreme deviations. Because its moments do not exist, common methods such as least squares can behave poorly under Cauchy-like noise. In such settings, robust estimators like the median or interquartile range provide more reliable summaries.

When analyzing data, it can be valuable to visualize how quickly probabilities accumulate in the Cauchy distribution. This calculator lets you experiment with different locations $x_0$, scales $\gamma$, and evaluation points $x$. Observe how the height of the density function decreases slowly compared with the normal distribution, and how the CDF transitions gradually from 0 to 1. Such experimentation builds intuition for heavy-tailed behavior.

Parameter Intuition

The location parameter $x_0$ marks the median and mode of the distribution, shifting the entire curve left or right without changing its shape. The scale $\gamma$ stretches the distribution horizontally; doubling $\gamma$ doubles the interquartile range and widens the central peak. Unlike the standard deviation in a normal distribution, $\gamma$ does not correspond to a finite variance, but it still provides a sense of spread.

Because the tails decay so slowly, outliers remain common no matter how narrow $\gamma$ becomes. This behavior serves as a reminder that the Cauchy distribution resists concentration: even with small scale, extreme values are inevitable.

Origins and Applications

The Cauchy distribution first appeared in Augustin-Louis Cauchy’s work on optical theory, where it described the intensity of light scattered by certain crystals. It later emerged in statistics as the ratio of two independent standard normal variables and as the distribution of the tangent of a uniformly distributed angle. Physicists encounter it when analyzing resonance phenomena, where energy builds up and decays around a central frequency, producing the characteristic peak and long tails.

In modern practice, the Cauchy distribution provides a heavy-tailed prior in Bayesian modeling, especially for parameters that might be large but are expected to be centered around zero. For example, using a Cauchy prior on regression coefficients imposes shrinkage yet allows occasional large effects. Signal processing engineers use the distribution to model impulsive noise, while navigation systems account for measurement errors that follow Cauchy-like behavior when sensors are susceptible to bursts of interference.

Simulation and Sampling

Generating random Cauchy values is as simple as drawing a uniform variable $U$ on (0,1) and computing $x_0+\gamma \times \tan \left(\pi ,(,U,-,1,/,2,)\right)$. Because this transformation involves the inverse CDF, it yields exact samples. Visualizing histograms of simulated values highlights how frequently observations stray far from the center, reinforcing the notion of heavy tails.

Monte Carlo simulation also offers a way to approximate probabilities. To estimate the chance that $X$ exceeds 10 when $x_0=0$ and $\gamma =1$, generate many samples and count how many exceed that threshold. The proportion should align with the survival function computed by this calculator, providing an empirical check on the formula.

Median and Quantiles

Because the Cauchy distribution lacks a finite mean, analysts often use the median to describe its center. For the standard Cauchy, both the median and mode occur at zero, aligning with the location parameter $x_0$. Quantiles are easy to compute from the inverse CDF. For example, the 75th percentile lies at $x_0+\gamma$, while the 25th percentile is one scale unit below the median. These wide-spaced quantiles reflect the distribution's propensity for extreme values.

Worked Example

Suppose a physics experiment records resonance frequencies with occasional large deviations. If you model the measurement error with a Cauchy distribution at $x_0=0$ and $\gamma =2$, an observation at $x=5$ has a density of $\frac{1}{\pi \times 2\times \left(1+{\frac{5}{2}}^2\right)}$. Although such a value is far from the median, it remains plausible because of the heavy tails. Running the calculator with these parameters lets you inspect both the PDF and CDF at that point and compare the likelihood to a normal model.

Estimating Parameters

Fitting a Cauchy distribution to data requires robust methods. The sample median is a consistent estimator for the location, while the half interquartile range estimates the scale. Maximum likelihood estimation is possible but more numerically challenging because the likelihood surface is flat near its maximum. Analysts often combine graphical tools like quantile plots with these simple estimators to judge how well the Cauchy model fits observations.

Common Pitfalls

The absence of finite moments means that sample means and variances are unreliable. Even after thousands of observations, the average of Cauchy data can wander arbitrarily far from $x_0$. Analysts should rely on medians and interquartile ranges instead of means and standard deviations. When fitting models, algorithms that assume quadratic loss—like ordinary least squares—may fail catastrophically under Cauchy errors; robust alternatives such as least absolute deviations fare better.

Another pitfall is misinterpreting the scale parameter as a standard deviation. Because $\gamma$ measures half the interquartile range, it cannot be compared directly to standard deviations from other distributions. Remembering this distinction prevents misleading comparisons between Cauchy and Gaussian models.

Further Reading

Heavy-tailed distributions such as the Cauchy appear in areas ranging from signal processing to finance. Texts on robust statistics offer strategies for handling such data, and many mathematical software packages provide built-in functions for the PDF, CDF, and quantile calculations. Experimenting with this calculator can serve as a springboard to those deeper explorations.