A C++ header-only library of statistical distribution functions.
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StatsLib   Mentioned in Awesome Cpp Build Status Coverage Status Codacy Badge License

StatsLib is a templated C++ library of statistical distribution functions, featuring unique compile-time computing capabilities and seamless integration with several popular linear algebra libraries.


  • A header-only library of probability density functions, cumulative distribution functions, quantile functions, and random sampling methods.
  • Functions are written in a specialized C++11 constexpr format, enabling the library to operate as both a compile-time and run-time computation engine.
  • Designed with a simple R-like syntax.
  • Optional vector-matrix functionality with wrappers to support:
  • Matrix-based operations are parallelizable with OpenMP.
  • Released under a permissive, non-GPL license.



Functions to compute the cdf, pdf, quantile, as well as random sampling methods, are available for the following distributions:

  • Bernoulli
  • Beta
  • Binomial
  • Cauchy
  • Chi-squared
  • Exponential
  • F
  • Gamma
  • Inverse-Gamma
  • Laplace
  • Logistic
  • Log-Normal
  • Normal (Gaussian)
  • Poisson
  • Student's t
  • Uniform
  • Weibull

In addition, pdf and random sampling functions are available for several multivariate distributions:

  • inverse-Wishart
  • Multivariate Normal
  • Wishart


StatsLib is a header-only library. Simply add the header files to your project using

#include "stats.hpp"

Jupyter Notebook

You can test the library online using an interactive Jupyter notebook:



The following options should be declared before including the StatsLib header files.

  • For inline-only functionality (i.e., no constexpr specifiers):
  • OpenMP functionality is enabled by default if the _OPENMP macro is detected (e.g., by invoking -fopenmp with a GCC or Clang compiler). To explicitly enable OpenMP features use:
  • To disable OpenMP functionality:
  • To use StatsLib with Armadillo, Blaze or Eigen:

Syntax and Examples

Functions are called using an R-like syntax. Some general rules:

  • density functions: stats::d*. For example, the Normal (Gaussian) density is called using
stats::dnorm(<value>,<mean parameter>,<standard deviation>);
  • cumulative distribution functions: stats::p*. For example, the Gamma CDF is called using
stats::pgamma(<value>,<shape parameter>,<scale parameter>);
  • quantile functions: stats::q*. For example, the Beta quantile is called using
stats::qbeta(<value>,<a parameter>,<b parameter>);
  • random sampling: stats::r*. For example, to generate a single draw from the Logistic distribution:
stats::rlogis(<location parameter>,<scale parameter>,<seed value or random number engine>);

All of these functions have matrix-based equivalents using Armadillo, Blaze, and Eigen dense matrices.

  • The pdf, cdf, and quantile functions can take matrix-valued arguments. For example,
// Using Armadillo:
arma::mat norm_pdf_vals = stats::dnorm(arma::ones(10,20),1.0,2.0);
  • The randomization functions (r*) can output random matrices of arbitrary size. For example,
// Armadillo:
arma::mat gamma_rvs = stats::rgamma<arma::mat>(100,50,3.0,2.0);
// Blaze:
blaze::DynamicMatrix<double> gamma_rvs = stats::rgamma<blaze::DynamicMatrix<double>>(100,50,3.0,2.0);
// Eigen:
Eigen::MatrixXd gamma_rvs = stats::rgamma<Eigen::MatrixXd>(100,50,3.0,2.0);

        will generate a 100-by-50 matrix of iid draws from a Gamma(3,2) distribution.

  • All matrix-based operations are parallelizable with OpenMP. For GCC and Clang compilers, simply include the -fopenmp option during compilation.


Random number seeding is available in two formats: seed values and random number engines.

  • Seed values are passed as unsigned integers. For example, to generate a draw from a normal distribution N(1,2) with seed value 1776:
  • Random engines in StatsLib use the 64-bit Mersenne-Twister generator (std::mt19937_64) and are passed by reference. Example:
std::mt19937_64 engine(1776);


More examples with code:

// evaluate the normal PDF at x = 1, mu = 0, sigma = 1
double dval_1 = stats::dnorm(1.0,0.0,1.0);
// evaluate the normal PDF at x = 1, mu = 0, sigma = 1, and return the log value
double dval_2 = stats::dnorm(1.0,0.0,1.0,true);
// evaluate the normal CDF at x = 1, mu = 0, sigma = 1
double pval = stats::pnorm(1.0,0.0,1.0);
// evaluate the Laplacian quantile at p = 0.1, mu = 0, sigma = 1
double qval = stats::qlaplace(0.1,0.0,1.0);

// draw from a t-distribution dof = 30
double rval = stats::rt(30);

// matrix output
arma::mat beta_rvs = stats::rbeta<arma::mat>(100,100,3.0,2.0);
// matrix input
arma::mat beta_cdf_vals = stats::pbeta(beta_rvs,3.0,2.0);

Compile-time Computation Capabilities

StatsLib is designed to operate equally well as a compile-time computation engine. Compile-time computation allows the compiler to replace function calls (e.g., dnorm(0,0,1)) with static values in the source code. That is, functions are evaluated during the compilation process, rather than at run-time. This capability is made possible due to the templated constexpr design of the library and can be verified by inspecting the assembly code generated by the compiler.

The compile-time features are enabled using the constexpr specifier. The example below computes the pdf, cdf, and quantile function of the Laplace distribution:

#include "stats.hpp"

int main()
    constexpr double dens_1  = stats::dlaplace(1.0,1.0,2.0); // answer = 0.25
    constexpr double prob_1  = stats::plaplace(1.0,1.0,2.0); // answer = 0.5
    constexpr double quant_1 = stats::qlaplace(0.1,1.0,2.0); // answer = -2.218875...

    return 0;

Assembly code generated by Clang without any optimization:

	.quad	-4611193153885729483    ## double -2.2188758248682015
	.quad	4602678819172646912     ## double 0.5
	.quad	4598175219545276417     ## double 0.25000000000000006
	.section	__TEXT,__text,regular,pure_instructions
	.globl	_main
	.p2align	4, 0x90
_main:                                  ## @main
	push	rbp
	mov	rbp, rsp
	xor	eax, eax
	movsd	xmm0, qword ptr [rip + LCPI0_0] ## xmm0 = mem[0],zero
	movsd	xmm1, qword ptr [rip + LCPI0_1] ## xmm1 = mem[0],zero
	movsd	xmm2, qword ptr [rip + LCPI0_2] ## xmm2 = mem[0],zero
	mov	dword ptr [rbp - 4], 0
	movsd	qword ptr [rbp - 16], xmm2
	movsd	qword ptr [rbp - 24], xmm1
	movsd	qword ptr [rbp - 32], xmm0
	pop	rbp


Keith O'Hara


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