Enter a location parameter (x₀), scale parameter (γ), and a value of x to compute the Cauchy distribution's PDF and CDF. You can also specify an interval [a, b] to get the probability that X falls between two values. Results include the probability density f(x), P(X < x), P(X > x), and P(a < X < b).
Probability Density f(x)
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P(X < x) — CDF
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P(X > x)
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P(a < X < b)
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P(X < a)
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P(X > b)
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The Cauchy distribution, also known as the Lorentz or Lorentzian distribution, is a continuous probability distribution that is symmetric and bell-shaped but with much heavier tails than the normal distribution. Unlike the normal distribution, the Cauchy distribution has no defined mean or variance, making it a popular example in probability theory and physics.
The location parameter (x₀ or t) specifies the peak of the distribution — the x-value where the PDF reaches its maximum. The scale parameter (γ or s) controls the spread or width of the distribution; larger values produce a wider, flatter curve. The scale parameter must always be greater than zero.
The probability density function (PDF) is given by f(x; x₀, γ) = 1 / (π · γ · [1 + ((x − x₀)/γ)²]). It reaches its maximum value of 1/(πγ) at x = x₀ and decreases symmetrically on both sides.
The cumulative distribution function (CDF) is F(x) = 1/2 + (1/π) · arctan((x − x₀)/γ). It gives the probability that the random variable X takes a value less than or equal to x. The CDF ranges from 0 to 1 and equals exactly 0.5 at x = x₀.
The Cauchy distribution's heavy tails cause the integrals that define the mean and variance to diverge — they do not converge to finite values. This makes the Cauchy distribution a classic example of a distribution where the sample mean does not converge to a stable value as sample size increases.
The interval probability P(a < X < b) is simply the difference of the CDF values: F(b) − F(a). Enter your lower bound (a) and upper bound (b) in the optional interval fields, and the calculator computes this automatically using the Cauchy CDF formula.
The Cauchy distribution appears frequently in physics to describe spectral line broadening (Lorentzian profile), in mechanical and electrical resonance systems, and in statistics as a robust example of a heavy-tailed distribution. It is also used in Bayesian analysis as a weakly informative prior for location parameters.
Yes. The Cauchy distribution, Lorentz distribution, Lorentzian function, and Breit-Wigner distribution are all names for the same probability distribution. The name 'Cauchy' is standard in mathematics and statistics, while 'Lorentzian' is more common in physics and spectroscopy.