StatMath.CdfFunctions

Cumulative Distribution Functions (CDF) This class provides static methods to calculate the cumulative distribution function for various statistical distributions.

The CDF, F(x), gives the probability that a random variable X will take a value less than or equal to x. Distribution Categories:

• Continuous distributions (Normal, Exponential, Gamma, Beta, etc.)

• Discrete distributions (Binomial, Poisson, Geometric, etc.)

• Special distributions (Chi-Square, F-distribution, Student’s t)

• Heavy-tailed distributions (Pareto, Weibull)

Calculates the CDF of a uniform distribution: F(x; a, b).

Usage

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

Functions

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

Calculates the CDF of a uniform distribution: F(x; a, b).

Returns the probability that a random variable from a uniform distribution on the interval [a, b] is less than or equal to x.

Mathematical Note: F(x) = (x-a)/(b-a) for a ≤ x ≤ b

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

Calculates the CDF of a normal distribution: F(x; μ, σ).

Returns the probability that a random variable from a normal (Gaussian) distribution with mean μ and standard deviation σ is less than or equal to x. Uses the erf() for computation.

Mathematical Note: F(x) = (1/2)[1 + erf((x-μ)/(σ√2))] See erf() for the error function implementation.

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

Calculates the CDF of an exponential distribution: F(x; λ).

Returns the probability that a random variable from an exponential distribution with rate parameter λ is less than or equal to x.

Mathematical Note: F(x) = 1 - e^(-λx) for x ≥ 0

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

Calculates the CDF of a beta distribution: F(x; α, β).

Returns the probability that a random variable from a beta distribution with shape parameters α and β is less than or equal to x. Uses the incomplete_beta().

Mathematical Note: F(x) = I_x(α, β) where I is the incomplete beta function. See incomplete_beta() for implementation details.

gamma_cdf(x: float, k_shape: float, theta_scale: float) → float: # Renamed k, theta

Calculates the CDF of a gamma distribution: F(x; k, θ).

Returns the probability that a random variable from a gamma distribution with shape parameter k and scale parameter θ is less than or equal to x. Uses the lower_incomplete_gamma_regularized().

Mathematical Note: F(x) = P(k, x/θ) where P is the regularized lower incomplete gamma function. See lower_incomplete_gamma_regularized() for implementation details.

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

Calculates the CDF of a chi-square distribution: F(x; k).

Returns the probability that a random variable from a chi-square distribution with k degrees of freedom is less than or equal to x. This is a special case of the gamma distribution.

Mathematical Note: Chi-square with k df is gamma_cdf(k/2, 2). See gamma_cdf() for the gamma CDF implementation.

f_cdf(x: float, d1_df: float, d2_df: float) → float: # Renamed d1, d2

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

Returns the probability that a random variable from an F-distribution with d1 and d2 degrees of freedom is less than or equal to x. Uses the regularized incomplete beta function.

Mathematical Note: F(x) = I_z(d1/2, d2/2) where z = d1x/(d1x + d2)

t_cdf(x_val: float, df_nu: float) → float: # Renamed x, df

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

Returns the probability that a random variable from a Student’s t-distribution with ν (nu) degrees of freedom is less than or equal to x. Uses the regularized incomplete beta function.

Mathematical Note: Uses transformation via incomplete_beta()

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

Calculates the CDF of a binomial distribution: F(k; n, p).

Returns the probability of observing k or fewer successes in n independent Bernoulli trials, each with success probability p. Computed as the sum of binomial_pmf() values.

Mathematical Note: F(k) = Σᵢ₌₀ᵏ (n choose i) p^i (1-p)^(n-i)

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

Calculates the CDF of a Poisson distribution: F(k; λ).

Returns the probability of observing k or fewer events in a fixed interval, given an average rate λ of events. Computed as the sum of Poisson PMF values. See poisson_pmf() for the PMF implementation.

Mathematical Note: F(k) = Σᵢ₌₀ᵏ (λ^i e^(-λ))/i!

geometric_cdf(k_trials: int, p_prob: float) → float:

Calculates the CDF of a geometric distribution: F(k; p).

Returns the probability that the first success in independent Bernoulli trials occurs on or before the k-th trial. Assumes k ≥ 1.

Mathematical Note: F(k) = 1 - (1-p)^k

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

Calculates the CDF of a negative binomial distribution: F(k; r, p).

Returns the probability that the r-th success occurs on or before the k-th trial in independent Bernoulli trials. Computed as the sum of negative_binomial_pmf() values.

Mathematical Note: F(k) = Σᵢ₌ᵣᵏ (i-1 choose r-1) p^r (1-p)^(i-r)

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

Calculates the CDF of a Pareto distribution: F(x; scale, shape).

Returns the probability that a random variable from a Pareto distribution with scale parameter (minimum value) and shape parameter is less than or equal to x. Uses the closed-form solution.

Mathematical Note: F(x) = 1 - (scale/x)^shape for x ≥ scale

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

Calculates the CDF of a Weibull distribution: F(x; λ, k).

Returns the probability that a random variable from a Weibull distribution with scale parameter λ and shape parameter k is less than or equal to x. Uses the closed-form solution. Widely used for reliability analysis.

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

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

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

Returns the probability that a random variable from a Cauchy distribution with location parameter x₀ and scale parameter γ is less than or equal to x. Uses the closed-form solution involving arctan.

Mathematical Note: F(x) = (1/π) * arctan((x - x₀)/γ) + 1/2

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

Calculates the CDF of a lognormal distribution: F(x; μ, σ).

Returns the probability that a random variable from a lognormal distribution with location parameter μ and scale parameter σ is less than or equal to x. If X ~ Lognormal(μ, σ), then ln(X) ~ Normal(μ, σ).

Mathematical Note: F(x) = Φ((ln(x) - μ)/σ) where Φ is the standard normal CDF