Cauchy Distribution
A heavy-tailed distribution where mean and variance are undefined. Educational for understanding pathological distributions.
Probability Density FunctionPDF = f(x)
Cumulative Distribution FunctionCDF = F(x)
Probability in an Interval
Compute the probability that X falls between two values. For continuous distributions, this is the shaded area under the PDF (≤ vs < doesn't matter).
P(a ≤ X ≤ b)
Quantiles (Inverse CDF)
Pick a probability p and read the corresponding quantile xₚ where F(xₚ) = p.
Quantile xₚ (where F(xₚ) = p)
In Cauchy Distribution, selected probability p: 0.9500 (95.00%)
Parameters
Adjust the parameters and (optionally) add up to 3 curves to compare different settings side‑by‑side.
0 curves shown
Estimating parameters from data
If you’re fitting this distribution to observed data, these are common plug‑in estimates you can start with.
Location (x₀)
Use the sample median (not the mean, which is undefined).
Scale (γ)
Estimate from the interquartile range: γ = IQR/2.