The standard Cauchy distribution is a Cauchy distribution with location parameter 0 and scale parameter 1. It arises naturally as the ratio of two independent standard normal random variables.
Probability Density Function
Support
Mean
Variance
| Example |
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| A line passes through the origin at a uniformly chosen random angle. The slope of the line is a standard Cauchy random variable. |
| Let X₁ and X₂ be independent standard normal random variables. Then X₁/X₂ is a standard Cauchy random variable. |
| A sample of size 2 with sample mean X̄ and sample variance S² is chosen from a normal distribution with mean μ. Then √2(X̄ − μ)/S is a standard Cauchy random variable. |
X ∼ Standard Cauchy
E(X) = , Var(X) =
Although the pdf of the standard Cauchy is similar to that of a standard normal distribution in being symmetric about the origin, a key feature of the standard Cauchy distribution is that neither the mean nor the variance is defined. This is a consequence of having fatter tails than the standard normal distribution. Because of this, the mean-variance box under the graph is not shown.
A key feature of the standard Cauchy distribution is that neither the mean nor the variance is defined. This is a consequence of the distribution having fatter tails than the standard normal distribution. Because of this, the mean-variance box under the graph is not shown.
The illustration above shows a red line passing through the point (0, 1), 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 standard Cauchy distribution.
The simulation above shows a red line passing through the point (0, 1), 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 standard Cauchy distribution. The histogram accumulates the results of each simulation.
Y =
E(Y) = , Var(Y) =