Inverse Gaussian (Wald) Distribution
X ~ InvGaussian(μ, λ)
First-Passage-Time Family • Published July 22, 2026
Defined by its probabilistic origin as the first-passage time of a drifting Brownian motion to a fixed level, and named for a cumulant-generating-function relationship to the already-published Normal rather than a literal reciprocal. A rare fully closed-form maximum-likelihood estimator, a confirmed Central-Limit-like convergence to the Normal as λ grows relative to μ, and simulation via the efficient Michael-Schucany-Haas algorithm verified against scipy.stats.invgauss to within 5.6e-17.
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About This Reference Sheet
Defined by its probabilistic origin as the first-passage time of a drifting Brownian motion to a fixed level, and named for a cumulant-generating-function relationship to the already-published Normal rather than a literal reciprocal. A rare fully closed-form maximum-likelihood estimator, a confirmed Central-Limit-like convergence to the Normal as λ grows relative to μ, and simulation via the efficient Michael-Schucany-Haas algorithm verified against scipy.stats.invgauss to within 5.6e-17.
Support
x ∈ (0, ∞)
Parameters
μ > 0 (mean), λ > 0 (shape)