Gompertz Distribution
X ~ Gompertz(b, η)
Mortality / Increasing-Hazard Family • Published July 22, 2026
Benjamin Gompertz's 1825 'law of human mortality': an exponentially increasing hazard function that remains the actuarial industry's standard adult-mortality model two centuries later. Defined directly by its hazard rather than its density, verified against scipy.stats.gompertz to within 7e-17, with no closed-form maximum-likelihood estimator and simulation via a closed-form inverse-CDF transform.
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About This Reference Sheet
Benjamin Gompertz's 1825 'law of human mortality': an exponentially increasing hazard function that remains the actuarial industry's standard adult-mortality model two centuries later. Defined directly by its hazard rather than its density, verified against scipy.stats.gompertz to within 7e-17, with no closed-form maximum-likelihood estimator and simulation via a closed-form inverse-CDF transform.
Support
x ∈ [0, ∞)
Parameters
b > 0 (shape, rate of hazard growth), η > 0 (scale)