Arctangent distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ arctan(λ,ϕ) is used to indicate that the random variable X has the arctangent distribution with phase shift parameter ϕ and positive location parameter λ. An arctangent random variable X with parameters λ and ϕ has probability density function

f(x) = \frac{λ}{(arctan(λϕ) + 1/2π) ( 1 + λ^{2}(x − ϕ)^{2})} x ≥ 0

for λ > 0 and −∞ < ϕ < ∞. The probability density function with three different choices of parameters is illustrated below.

PSfrag replacements λ = 1, ϕ = 1

λ = 2, ϕ = 3
λ = 1, ϕ = 3
x
f(x)

The cumulative distribution function of X is

F (x) = P(X ≤ x) = 2( \frac{arctan(λϕ) − arctan(−xλ + λϕ)}{2 arctan(λϕ) + π} ) x ≥ 0.

The survivor function of X is

S(x) = P(X ≥ x) = \frac{π + 2 arctan(−xλ + λϕ)}{2 arctan(λϕ) + π} x ≥ 0.

The hazard function of X is

h(x) = \frac{f(x)}{S(x)} = \frac{2λ}{(1 + λ^{2}x^{2}− 2λ^{2}ϕx + λ^{2}ϕ^{2})(π + 2 arctan(−xλ + λϕ) )} x ≥ 0.

The cumulative hazard function of X is

H(x) = ln(2 arctan(λϕ) + π) − ln(π + 2 arctan(λ (−x + ϕ) )) x ≥ 0.

The inverse distribution function of X is

F^{−1} (u) = \frac{λϕ + tan(−arctan(λϕ) + uarctan(λϕ) + 1/2uπ)}{λ} 0 < u < 1.

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 := ArcTanRV(lambda, phi); CDF(X); SF(X); HF(X); IDF(X); Mean(X); Variance(X); Skewness(X); Kurtosis(X); MGF(X);

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