Principle:Avhz RustQuant Probability Distributions

Knowledge Sources	RustQuant Math Distributions
Domains	Statistics, Probability_Theory
Last Updated	2026-02-07 21:00 GMT

Overview

Probability distribution theory providing PDF, CDF, moment generation, and sampling for common continuous and discrete distributions.

Description

Probability Distributions in RustQuant are built around a core Distribution trait that defines a uniform interface for all supported distributions. Every distribution must implement the following methods:

• cf -- Characteristic function: $\phi (t)=E[e^{itX}]$, returning a Complex<f64>.
• pdf -- Probability density function (for continuous distributions).
• pmf -- Probability mass function (for discrete distributions).
• cdf -- Cumulative distribution function.
• inv_cdf -- Inverse CDF (quantile function).
• mean, median, mode -- Central tendency measures.
• variance, skewness, kurtosis -- Spread and shape measures.
• entropy -- Shannon entropy of the distribution.
• mgf -- Moment generating function: $M(t)=E[e^{tX}]$.
• sample -- Random variate generation, returning Result<Vec<f64>, RustQuantError>.

The library also defines a DistributionClass enum distinguishing between Discrete and Continuous distributions, and provides a standard-normal convenience constant N for use in option pricing.

The following distributions are implemented:

• Gaussian (Normal): $X\sim N(\mu ,{\sigma }^2)$
• Bernoulli: $X\sim \text{Bern}(p)$
• Binomial: $X\sim B(n,p)$
• Poisson: $X\sim \text{Pois}(\lambda )$
• Uniform: $X\sim U(a,b)$
• Exponential: $X\sim \text{Exp}(\lambda )$
• Gamma: $X\sim \mathrm{\Gamma }(\alpha ,\beta )$
• Chi-Squared: $X\sim {\chi }^2(k)$

Usage

Use the Probability Distributions principle when computing option prices (via the standard normal CDF), generating Monte Carlo simulations (via the sample method), performing statistical analysis, or modeling stochastic financial processes. The trait-based design allows generic algorithms to operate over any distribution type.

Theoretical Basis

For the Gaussian distribution, the PDF is:

$f(x)=\frac{1}{\sqrt{2\pi {\sigma }^2}}\exp (-\frac{(x-\mu {)}^2}{2{\sigma }^2})$

The CDF is computed using the complementary error function to avoid subtractive cancellation in the tails:

$\mathrm{\Phi }(x)=\frac{1}{2}\mathrm{erfc}(-\frac{x-\mu }{\sqrt{2}\sigma })$

The characteristic function is:

$\phi (t)=\exp (i\mu t-\frac{{\sigma }^2{t}^2}{2})$

The moment generating function is:

$M(t)=\exp (\mu t+\frac{{\sigma }^2{t}^2}{2})$

Random sampling uses the rand_distr crate for efficient variate generation.

Related Pages

Implemented By

• Implementation:Avhz_RustQuant_Gaussian_Distribution
• Implementation:Avhz_RustQuant_Distribution_Trait
• Implementation:Avhz_RustQuant_Uniform_Distribution
• Implementation:Avhz_RustQuant_Chi_Squared_Distribution
• Implementation:Avhz_RustQuant_Poisson_Distribution
• Implementation:Avhz_RustQuant_Binomial_Distribution
• Implementation:Avhz_RustQuant_Bernoulli_Distribution
• Implementation:Avhz_RustQuant_Exponential_Distribution
• Implementation:Avhz_RustQuant_Gamma_Distribution