The Cauchy distribution is a continuous probability distribution known for its heavy tails and undefined mean and variance. Its shape resembles a bell curve like the normal distribution but has much thicker tails and a sharp peak at the center. It is defined by two parameters: a location parameter (x₀), which indicates the peak, and a scale parameter (γ), which controls the spread.
Because of its heavy tails, the Cauchy distribution is used in contexts where extreme outliers are more common and standard deviation is not a useful measure. It serves as a cautionary example in statistics because the mean and variance are mathematically undefined, which makes common statistical methods inappropriate.
For example, the Cauchy distribution can model resonance behavior in physics, where a system may oscillate with large and unpredictable magnitudes near its natural frequency.
The graph below displays two Cauchy distributions with different location parameters (spread).
