The Cauchy distribution is a symmetric bell-shaped distribution which arises naturally as the ratio of two independent normal random variables with mean zero. The median is given by a parameter θ, while a scale parameter σ controls the spread of the distribution.
| Parameter | Range | Description |
|---|---|---|
| θ | −∞ < θ < ∞ | Location parameter |
| σ | σ > 0 | Scale parameter |
Probability Density Function
Support
Mean
Variance
| Example | θ | σ |
|---|---|---|
| Let X₁ and X₂ be independent standard normal random variables. Then X₁/X₂ is a Cauchy(0, 1) random variable. | 0.000 | 1.000 |
| Let X₁ be a normal(0, 6) and X₂ an independent normal(0, 2) random variables. Then X₁/X₂ is a Cauchy(0, 3) random variable. | 0.000 | 3.000 |
| A line passes through the point (1, 2) at a uniformly random angle. The x-axis intercept of the line is a Cauchy(1, 2) random variable. | 1.000 | 2.000 |
X ∼ Cauchy(θ, σ)
E(X) = , Var(X) =
Although the pdf of the Cauchy is similar to that of a normal distribution in being symmetric about θ, the Cauchy distribution is heavy-tailed, with neither the mean nor the variance being defined. Because of this, the mean-variance box under the graph is not shown.
Although the cdf of the Cauchy looks similar to that of a normal distribution, the Cauchy distribution is heavy-tailed, with neither the mean nor the variance being defined. Because of this, the mean-variance box under the graph is not shown.
The illustration above shows a red line passing through the point (θ, σ), where the angle of the line is uniformly random. The light blue circle shows the location X of the x-axis intercept of this line, which has a Cauchy(θ, σ) distribution.
The simulation above shows a red line passing through the point (θ, σ), where the angle of the line is uniformly random. The light blue circle shows the location X of the x-axis intercept of this line, which has a Cauchy(θ, σ) distribution. The histogram accumulates the results of each simulation.
Y =
E(Y) = , Var(Y) =