Inputs
About this distribution
The Cauchy distribution with location x₀ and scale γ > 0 has density
f(x) = [1/(πγ)] · [1 / (1 + ((x − x₀)/γ)²)]
and cumulative distribution
F(x) = (1/π) · arctan((x − x₀)/γ) + 1/2.
Use this page to compute F(x), the survival S(x) = 1 − F(x), and interval mass P(a < X ≤ b).
Input tips
- Default standard case is x₀ = 0 and γ = 1.
- γ must be positive; smaller γ gives a sharper peak and heavier center.
- To approximate ±∞ for x, use very large magnitude values.
- Increase decimal places for more detailed outputs.
Frequently Asked Questions
1) What does the CDF represent for this distribution?
The CDF at x gives the probability that a random draw from the Cauchy distribution is less than or equal to x, given the chosen location x₀ and scale γ.
2) How is the CDF computed here?
It uses F(x) = (1/π)·arctan((x − x₀)/γ) + 1/2 with a numerically stable atan2 implementation and double-precision floating point arithmetic.
3) What do x₀ and γ mean?
x₀ is the location (the median and mode), shifting the distribution along the x-axis. γ is the scale; larger γ spreads the distribution wider, smaller γ concentrates it near x₀.
4) Why are mean and variance not defined?
The tails of the Cauchy distribution are heavy enough that the integrals for mean and variance diverge; these moments do not exist.
5) Can I compute interval probabilities?
Yes. Provide a and b to get P(a < X ≤ b) = F(b) − F(a). This is useful for central coverage or one-sided bounds.
6) Is the survival function available?
The survival function S(x) = 1 − F(x) is reported alongside the CDF, representing the probability that X exceeds x.
7) What numerical precision is used?
Results are printed to a user-selected number of decimal places (0–18). Internally, calculations use standard double precision with stable trigonometric functions.