Module special 

Special mathematical functions.

Numerical approximations of standard mathematical functions used throughout probability and statistics.

Functions

chi_squared_cdf

CDF of the chi-squared distribution: P(X ≤ x | k).

erf

Error function erf(x).

erfc

Complementary error function erfc(x) = 1 − erf(x).

f_distribution_cdf

CDF of the F-distribution: P(X ≤ x | df1, df2).

f_distribution_quantile

Quantile function (inverse CDF) of the F-distribution.

gamma

Gamma function Γ(x) = exp(ln_gamma(x)).

inverse_normal_cdf

Approximation of the inverse standard normal CDF (quantile function).

ln_beta

Log of the Beta function: ln B(a, b) = ln Γ(a) + ln Γ(b) − ln Γ(a+b).

ln_gamma

Lanczos approximation of ln Γ(x).

regularized_incomplete_beta

Regularized incomplete beta function I_x(a, b).

regularized_lower_gamma

Regularized lower incomplete gamma function P(a, x) = γ(a, x) / Γ(a).

standard_normal_cdf

Approximation of the standard normal CDF Φ(x) = P(Z ≤ x) for Z ~ N(0,1).

standard_normal_pdf

Standard normal PDF φ(x) = (1/√(2π)) exp(-x²/2).

t_distribution_cdf

CDF of Student’s t-distribution: P(T ≤ t | df).

t_distribution_pdf

PDF of Student’s t-distribution.

t_distribution_quantile

Quantile function (inverse CDF) of Student’s t-distribution.