StatMath.PmfPdfFunctions

Probability Mass Functions (PMF) and Probability Density Functions (PDF) This class provides static methods to calculate probability mass functions for discrete distributions and probability density functions for continuous distributions.

The PMF/PDF gives the probability (or density) of a random variable taking on a specific value. Distribution Categories:

• PMF for discrete distributions (Binomial, Poisson, Negative Binomial)

• PDF for continuous distributions (Normal, Exponential, Uniform, Gamma, Beta, Chi-squared, Student’s t, F-distribution)

• Uses logarithmic calculations for numerical stability

Calculates the PMF of a binomial distribution: P(X = k | n, p). Returns the probability of observing exactly k successes in n

Usage

# Access via StatMath singleton
var result = StatMath.PmfPdfFunctions.function_name(parameters)

Functions

binomial_pmf(k_successes: int, n_trials: int, p_prob: float) → float:

Calculates the PMF of a binomial distribution: P(X = k | n, p).

Returns the probability of observing exactly k successes in n independent Bernoulli trials, each with success probability p. Uses logarithmic calculations for numerical stability.

Mathematical Note: P(X = k) = (n choose k) p^k (1-p)^(n-k)

poisson_pmf(k_events: int, lambda_param: float) → float:

Calculates the PMF of a Poisson distribution: P(X = k | λ).

Returns the probability of observing exactly k events in a fixed interval, given an average rate λ of events. Uses logarithmic calculations for numerical stability.

Mathematical Note: P(X = k) = (λ^k e^(-λ)) / k!

negative_binomial_pmf(k_trials: int, r_successes: int, p_prob: float) → float:

Calculates the PMF of a negative binomial distribution: P(X = k | r, p).

Returns the probability that the r-th success occurs on exactly the k-th trial in independent Bernoulli trials with success probability p. Uses logarithmic calculations for numerical stability.

Mathematical Note: P(X = k) = (k-1 choose r-1) p^r (1-p)^(k-r)

geometric_pmf(k_trial: int, p_prob: float) → float:

Calculates the PMF of a geometric distribution: P(X = k | p).

Returns the probability that the first success occurs on exactly the k-th trial in independent Bernoulli trials with success probability p. Uses standard parameterization where k ≥ 1.

Mathematical Note: P(X = k) = (1-p)^(k-1) * p for k ≥ 1

normal_pdf(x: float, mu: float = 0.0, sigma: float = 1.0) → float:

Calculates the PDF of a normal distribution: f(x; μ, σ).

Returns the probability density at x for a normal (Gaussian) distribution with mean μ and standard deviation σ.

Mathematical Note: f(x) = (1/σ√(2π)) e^(-(x-μ)²/(2σ²))

exponential_pdf(x: float, lambda_param: float) → float:

Calculates the PDF of an exponential distribution: f(x; λ).

Returns the probability density at x for an exponential distribution with rate parameter λ. Used for modeling waiting times and decay processes.

Mathematical Note: f(x) = λe^(-λx) for x ≥ 0, 0 otherwise

uniform_pdf(x: float, a: float, b: float) → float:

Calculates the PDF of a uniform distribution: f(x; a, b).

Returns the probability density at x for a uniform distribution on the interval [a, b].

Mathematical Note: f(x) = 1/(b-a) for a ≤ x ≤ b, 0 otherwise

gamma_pdf(x: float, k_shape: float, theta_scale: float) → float:

Calculates the PDF of a gamma distribution: f(x; k, θ).

Returns the probability density at x for a gamma distribution with shape parameter k and scale parameter θ.

Mathematical Note: f(x) = (1/(Γ(k)θ^k)) x^(k-1) e^(-x/θ) for x ≥ 0

beta_pdf(x: float, alpha: float, beta_param: float) → float:

Calculates the PDF of a beta distribution: f(x; α, β).

Returns the probability density at x for a beta distribution with shape parameters α and β. Defined on [0, 1].

Mathematical Note: f(x) = (Γ(α+β)/(Γ(α)Γ(β))) x^(α-1) (1-x)^(β-1)

weibull_pdf(x: float, scale_param: float, shape_param: float) → float:

Calculates the PDF of a Weibull distribution: f(x; λ, k).

Returns the probability density at x for a Weibull distribution with scale parameter λ and shape parameter k. Widely used in reliability analysis, survival analysis, and failure modeling.

Mathematical Note: f(x) = (k/λ)(x/λ)^(k-1) e^(-(x/λ)^k) for x ≥ 0

lognormal_pdf(x: float, mu: float, sigma: float) → float:

Calculates the PDF of a lognormal distribution: f(x; μ, σ).

Returns the probability density at x for a lognormal distribution with location parameter μ and scale parameter σ. If X ~ Lognormal(μ, σ), then ln(X) ~ Normal(μ, σ).

Mathematical Note: f(x) = (1/(xσ√(2π))) e^(-((ln(x)-μ)²)/(2σ²)) for x > 0

chi_squared_pdf(x: float, k_df: float) → float:

Calculates the PDF of a chi-squared distribution: f(x; k).

Returns the probability density at x for a chi-squared distribution with k degrees of freedom. This is a special case of the gamma PDF. See gamma_pdf() for the general gamma PDF implementation.

Mathematical Note: f(x) = (1/(2^(k/2)Γ(k/2))) x^(k/2-1) e^(-x/2) for x ≥ 0

t_pdf(x: float, df_nu: float) → float:

Calculates the PDF of a Student’s t-distribution: f(x; ν).

Returns the probability density at x for a Student’s t-distribution with ν (nu) degrees of freedom.

Mathematical Note: f(x) = (Γ((ν+1)/2)/(√(νπ)Γ(ν/2))) (1 + x²/ν)^(-(ν+1)/2)

f_pdf(x: float, d1_df: float, d2_df: float) → float:

Calculates the PDF of an F-distribution: f(x; d1, d2).

Returns the probability density at x for an F-distribution with numerator degrees of freedom d1 and denominator degrees of freedom d2.

Mathematical Note: Uses beta_function() relationship for numerical stability

cauchy_pdf(x: float, location: float = 0.0, scale: float = 1.0) → float:

Calculates the PDF of a Cauchy (Lorentzian) distribution: f(x; x₀, γ).

Returns the probability density at x for a Cauchy distribution with location parameter x₀ and scale parameter γ. The Cauchy distribution has undefined mean and variance due to heavy tails.

Mathematical Note: f(x) = 1/(πγ(1 + ((x-x₀)/γ)²))

pareto_pdf(x: float, scale_param: float, shape_param: float) → float:

Calculates the PDF of a Pareto distribution: f(x; xₘ, α).

Returns the probability density at x for a Pareto distribution with scale parameter xₘ and shape parameter α. Used to model the “80/20 rule” and power-law distributions.

Mathematical Note: f(x) = (α×xₘᵅ)/x^(α+1) for x ≥ xₘ