normal.md

Normal Distribution

Distribution index · Generalized Normal · Mixed-data likelihood

Purpose and Parameterization

The Normal model is a symmetric location-scale distribution for an unconstrained response. RMC.Numerics uses constructor order $(\mu,\sigma)$, where mean $\mu\in\mathbb R$ and standard deviation $\sigma>0$ have observation units. Direct Normal modeling of discharge assigns positive probability to negative values; that may be negligible for some data but is a structural assumption that must be checked.

Distribution

With $z=(x-\mu)/\sigma$, standard Normal density $\phi$, and CDF $\Phi$,

$$ f(x\mid\mu,\sigma)=\frac{1}{\sigma}\phi(z) =\frac{1}{\sigma\sqrt{2\pi}}e^{-z^2/2},\qquad F(x)=\Phi(z). \tag{NOR.1} $$

The survival and quantile functions are

$$ S(x)=1-\Phi(z),\qquad Q(p)=\mu+\sigma\Phi^{-1}(p). \tag{NOR.2} $$

Mean, median, and mode equal $\mu$; variance is $\sigma^2$, skewness is zero, and ordinary kurtosis is $3$. All moments exist and both tails are unbounded with Gaussian decay.

Likelihood and Posterior

For exact independent observations,

$$ \ell(\mu,\sigma)=-n\log\sigma-\frac{n}{2}\log(2\pi) -\frac{1}{2\sigma^2}\sum_i(x_i-\mu)^2. \tag{NOR.3} $$

The RMC.BestFit mixed-data likelihood uses $f$, $F$, and $S$ from (NOR.1)–(NOR.2) for exact, censored, uncertain, interval, and threshold information. Under nonstationarity, $\mu_i$ and/or $\sigma_i$ are trend-model predictions. The posterior combines every likelihood component with configured parameter and quantile priors. RMC.BestFit derives sample-dependent initialization and working bounds through the Numerics constraint routine; the mathematical constraints remain $\mu\in\mathbb R$ and $\sigma>0$.

The likelihood can drive $\sigma$ toward zero for degenerate data, so repeated or nearly identical observations require careful validation. A Normal fit can be useful for transformed data or approximately symmetric residuals, but direct rare-flood extrapolation is sensitive to the unbounded lower support and light upper tail.

Compile-Checked API Example

private static (double Density, double Cdf, double Quantile) EvaluateNormal()
{
    var distribution = new global::Numerics.Distributions.Normal(1500.0, 400.0);
    return Evaluate(distribution, value: 2200.0, nonExceedanceProbability: 0.99);
}

All magnitudes use the same units. If the modeled quantity is annual maximum discharge, the analyst must still justify approximate independence, stationarity or covariate trends, and the Normal tail.

Numerical Implementation and Validation

CDF and inverse-CDF calculations use the corresponding Numerics special-function paths. Required checks include $F(Q(p))\approx p$, symmetry, known moments, normalized density, and convergence of Pearson III and generalized Normal calculations as skew/shape approaches zero. The API example is compile-checked but intentionally reports no hand-written numerical output.

The distribution verification matrix identifies the current independent formula and fitted-objective comparisons, retained Bayesian/MLE recovery designs, and their distinct acceptance rules. The compiled example guards API compatibility; it does not run an estimator or establish scientific accuracy by itself.

References

[1] N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, vol. 1, 2nd ed. Wiley, 1994.