Hyperbolic-secant distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The hyperbolic-secant distribution has probability density function

f(x) = sech(πx), −∞ < x < ∞,

where the hyperbolic-secant function is defined by

sech(z) = \frac{2}{e^{z}+ e^{−z}}

for −∞ < z < ∞. 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 + 2arctan(sinh(πx))}{2π} − ∞ < x < ∞.

The survivor function on the support of X is

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

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = −\frac{2π}{cosh(πx)(π − 2arctan(sinh(πx)))} − ∞ < x < ∞.

The cumulative hazard function on the support of X is

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

The inverse distribution function of X is

F^{−1} (u) = \frac{arcsin(cot(πu))}{π} 0 < u < 1.

The median of X is 0.

The moment generating function of X is

M(t) = E [e^{tX}] = ∫_{−∞}^{∞} sech(πx) t > 0.

The characteristic function of X is

ϕ(t) = E [e^{itX}] = ∫_{−∞}^{∞} sech(πx) t > 0.

The population mean, variance, skewness, and kurtosis of X are

E[X] = 0  V [X] = 1  E [( \frac{X − μ}{σ} ) ^{3}] = 0  E [( \frac{X − μ}{σ} ) ^{4}] = 2.

APPL verification: The APPL statements

X := HyperbolicSecantRV(); CDF(X); SF(X); HF(X); CHF(X); IDF(X); Mean(X); Skewness(X); MGF(X);

verify the cumulative distribution function, survivor function,hazard function, cumulative hazard function, inverse function, population mean, skewness, and moment generating function. APPL fails to verify the population variance and kurtosis.