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Cauchy Distribution

A real-line probability law with a rational density, heavy tails, undefined mean and variance, and an iid sample average that retains the original law.

Version
v1 · 2026-10-31 · History
Domain-specific #
14470
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomain
Probability Theory → Mathematics
Aliases
Lorentz distribution

Core Idea

A Cauchy distribution is a probability law on the real line with location \(\mu\) and scale \(\gamma>0\). Its density is \(f(y)=\gamma/[\pi((y-\mu)^2+\gamma^2)]\). The center is both median and mode; the usual mean and variance do not exist.[1]

The arithmetic average of independent, identically distributed draws from the same Cauchy law has that same law. A finite sample mean can still be calculated, but it does not concentrate around a finite population mean as the sample grows.[1][2]

How would you explain it like I'm…

The Wild-Spinner Pattern

Imagine a magic spinner that picks numbers. Most of the time it lands near the middle, but every now and then it lands super, super far away. Those far-away ones happen often enough that averaging lots of spins never settles down. So to find the middle, it's better to line up the results and pick the one in the middle.

Averages That Never Settle

The Cauchy distribution is a rule in math for how likely different numbers are when something random happens. Its picture is a hill with a center and a width you can change. But the sides of the hill fade away slowly, so really huge numbers keep popping up now and then. Because of that, if you take the average of many results, it jumps around just as much as one single result — it never settles down to a fixed answer. The middle-most result, called the median, does a much better job of finding the center.

Heavy-Tailed Law Without a Mean

The Cauchy distribution is a probability distribution over all real numbers, with a location μ (its center) and a positive scale γ (its width). Its density is f(y) = γ / [π((y−μ)² + γ²)], which integrates to 1, so it truly describes a random number, not just a bell-like curve. Its tails shrink only like 1/y², so slowly that the mean and variance are undefined; μ is the median and the most likely value, not an average. Surprisingly, the average of many independent Cauchy draws with the same μ and γ has exactly the same distribution as a single draw, so bigger samples don't make the mean more precise. That's why the median is a better way to estimate the center here. A physics line shape (a Lorentzian) can look like this curve without being a probability distribution.

 

The Cauchy distribution is a continuous probability law on the real line with location parameter μ and scale parameter γ > 0, density f(y) = γ / [π((y−μ)² + γ²)]. Shifting μ moves the center; changing γ changes the width; the density integrates to one, so it is a genuine law for a real-valued random variable. Its tails decay like 1/y², which makes the expectation and variance undefined; μ is the median and mode rather than an expected value. For i.i.d. Cauchy draws with common location and scale, the sample mean has the same Cauchy law as a single observation, so the mean does not concentrate as the sample size grows — the usual law-of-large-numbers intuition fails. NIST illustrates this with a simulated sample whose mean was 3.70 while its median was −0.016, showing why the median is the more useful location summary. The name refers to the exact normalized probability object; a Lorentzian intensity profile in physics may resemble the density, but proportional shape alone does not assert a probability law for a random wavelength.

Structural Signature

  • Outcome: a real-valued random coordinate on the full real line.
  • Parameters: location \(\mu\) and positive scale \(\gamma\) determine center and width.
  • Law: the normalized rational density allocates probability over that outcome space.
  • Consequences: heavy tails, undefined moments and stability of iid averages follow from the law; they do not require a particular generating process.[1][2]

What It Is Not

A heavy-tailed law or an approximately Lorentzian intensity curve is not automatically the exact normalized Cauchy probability law. A reflected half-Cauchy law has only positive support and a different normalization.[3][4]

Scope of Application

The exact law can describe simulated samples or a mathematical first-hit coordinate. A plausible curve shape in physical data alone does not prove that a random outcome has this distribution.[5][4][3]

Clarity

A sample mean is computable from any finite sample even though the theoretical Cauchy expectation is undefined. NIST reports a simulated sample mean of 3.70 and median of −0.016, illustrating why one realization's mean need not describe its center.[5]

Examples

Simulated sample. NIST describes 10,000 generated Cauchy draws and reports a mean and median for that particular realization. The density and stability are properties of the law, not claims that every sample reproduces those numbers.[5][1]

Brownian first hit. For planar Brownian motion starting distance \(x>0\) from the \(y\)-axis, the first hit's \(y\) coordinate has density \(x/[\pi(x^2+y^2)]\) on the full real line: a Cauchy law with location zero and scale \(x\). The paper's later reflected quadrant process is a different, positive-half-line case.[4]

Reasoning and Boundary

Identify the random outcome, its support and normalization, then test the exact density before transferring moment or averaging conclusions. Probability Distribution is a broader mathematical comparison; the named law specifies one family. No graph edge is implied by that comparison.[1]

Neighborhood in Abstraction Space

Cauchy Distribution sits in a sparse region of the domain-specific corpus (95th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2835 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-31

References

[1] National Institute of Standards and Technology, “Cauchy Distribution,” NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.3, https://itl.nist.gov/div898/handbook/eda/section3/eda3663.htm (density and distribution properties). registry ↩a ↩b ↩c ↩d ↩e

[2] Giovanni Puccetti, Pietro Rigo, Bin Wang, and Ruodu Wang, “Centers of probability measures without the mean,” author-hosted full manuscript (2018), https://sas.uwaterloo.ca/~wang/papers/2018Puccetti-Rigo-Wang-Wang-JTP.pdf (printed p.2 / one-based PDF p.3 gives standard-normal-ratio, stability, and ray constructions; printed p.5 / one-based PDF p.6 gives standard Cauchy density). These are mathematical background facts in the article, not reported experiments. registry ↩a ↩b

[3] National Institute of Standards and Technology, “Spectral Line Shapes, Widths, and Shifts,” in Atomic Spectroscopy, https://physics.nist.gov/Pubs/AtSpec/node20.html (official HTML inspected; Pressure Broadening passage says intensity profiles are often approximately Lorentzian and gives a proportional profile, not an exact random-wavelength Cauchy probability assertion). registry ↩a ↩b

[4] Jonas Arista and Neil O’Connell, “Loop-Erased Walks and Random Matrices,” Journal of Statistical Physics 177 (2019), 528–567, DOI 10.1007/s10955-019-02378-1, https://link.springer.com/article/10.1007/s10955-019-02378-1 (original mathematical research, publisher full HTML inspected; §4.3 gives full-line ordinary Brownian \(h'\) immediately before Equation (4.7), then reflected positive-quadrant \(h\) in Equation (4.7)). registry ↩a ↩b ↩c

[5] National Institute of Standards and Technology, “Measures of Location,” NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.5.1, https://itl.nist.gov/div898/handbook/eda/section3/eda351.htm (simulated-sample and location discussion). registry ↩a ↩b ↩c