Standard Cauchy distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ Cauchy(1,0) is used to indicate that the random variable X has the standard Cauchy distribution. A standard Cauchy random variable X has probability density function

f(x) = \frac{1}{π (1 + x^{2})} − ∞ < x < ∞.

The probability density function is illustrated below.

PSfrag replacements

x
f(x)

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = \frac{π + 2 arctan(x)}{2π} − ∞ < x < ∞.

The survivor function on the support of X is

S(x) = P(X ≥ x) = \frac{π − 2 arctan(x)}{2π} − ∞ < x < ∞.

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = \frac{2}{(1 + x^{2}) (π − 2 arctan(x))} − ∞ < x < ∞.

The cumulative hazard function on the support of X is

H(x) = −lnS(x) = ln(2)+ ln(π)− ln(π − 2 arctan(x)) − ∞ < x < ∞.

The inverse distribution function of X is

F^{−1} (u) = −cot(u · π) 0 < u < 1.

The median of X is 0.

The moments of X are undefined. It follows that the population mean, variance, skewness, and kurtosis of X are also undefined.

APPL verification: The APPL statements

X := StandardCauchyRV( ); CDF(X); SF(X); HF(X); CHF(X); IDF(X);

verify the cumulative distribution function, survivor function, hazard function, and cumulative hazard function.