4. Probability and statistics foundations for UQ

Uncertainty quantification (UQ) for occupant injury prediction rests on a simple premise: every quantity we care about — peak head acceleration, HIC₁₅, chest deflection, femur force, AIS severity — is treated as a random variable rather than a single deterministic number. A finite element (FE) crash model with a nominal input set produces one response value; a UQ workflow acknowledges that material properties, restraint timing, occupant posture, and impact conditions vary, and it asks how the response is distributed as a consequence. This chapter establishes the probabilistic vocabulary and the specific distributional choices that recur throughout occupant safety analysis.

Random variables and their two flavors

A random variable (RV) is a measurable function mapping outcomes in a sample space to real numbers. In occupant injury work we encounter both types constantly. Continuous RVs include HIC₁₅, peak chest acceleration (g), chest deflection (mm), and femur axial force (kN) — quantities that can, in principle, take any value in an interval. Discrete RVs include AIS level (0–6), airbag deployment stage (single-stage, dual-stage, adaptive), and the count of rib fractures. The distinction matters because it dictates which distribution family, density function, and estimation procedure apply.

Each RV is characterized by a cumulative distribution function (CDF), F(x) = P(X ≤ x), and — for continuous variables — a probability density function (PDF), f(x) = dF/dx. In injury prediction, the CDF is often the more useful object: the probability that HIC₁₅ exceeds a regulatory or injury-reference threshold is simply 1 − F(threshold).

Expectation and variance: the first two moments

The expectation E[X] (mean, μ) is the probability-weighted average of the RV. For a continuous response it is E[X] = ∫ x f(x) dx; for a discrete response, E[X] = Σ x_i p(x_i). The variance Var(X) = E[(X − μ)²] = E[X²] − μ² measures dispersion around the mean, and its square root σ is the standard deviation in the same units as the response — far more interpretable when reporting, say, chest deflection in millimetres.

Two properties are used constantly in UQ propagation. First, expectation is linear: E[aX + bY] = aE[X] + bE[Y] regardless of dependence. Second, variance is not linear in general: Var(aX + bY) = a²Var(X) + b²Var(Y) + 2ab·Cov(X,Y). The covariance term is the reason correlated inputs — for example, belt stiffness and belt pretensioner timing that are tuned together — can produce response variance far larger than a naive independent-input assumption would suggest. Higher moments (skewness, kurtosis) matter when injury responses are strongly asymmetric or heavy-tailed, which is common for severe-injury tails.

Common injury response distributions

Choosing a distribution family is a modelling decision with physical consequences. The table below summarizes the families most frequently used for occupant injury responses, with the rationale for each.

For HIC₁₅ specifically, the lognormal is a defensible default: HIC is strictly positive, its distribution across a population of crash pulses is right-skewed, and log-transforming it yields an approximately normal variable suitable for standard statistical inference. A useful practical check is to plot the empirical CDF of HIC on lognormal probability paper; if the points fall near a straight line, the lognormal assumption is reasonable. If the tail bends upward, a Weibull or a mixture may be warranted.

From distributions to injury risk: the tolerance curve

Injury criteria are frequently expressed as risk functions. A common form is the lognormal risk curve, P(injury) = Φ[(ln(x) − μ)/σ], where Φ is the standard normal CDF and μ, σ are fitted to biomechanical tolerance data. This is the same lognormal family introduced above, now used as a risk model rather than a response model. The distinction is important: the response distribution describes what the model predicts; the risk curve describes the probability of injury given a response value. UQ combines the two by integrating the response distribution against the risk curve — a step developed in later chapters but whose foundation is the CDF machinery introduced here.

Estimating moments from simulation output

In practice, the response distribution is not known analytically; it is estimated from a sample of simulation runs. Given N runs with HIC values x₁,…,x_N, the sample mean and variance are:

mu_hat    = (1/N) * sum(x_i)
var_hat   = (1/(N-1)) * sum((x_i - mu_hat)^2)
sigma_hat = sqrt(var_hat)
cv        = sigma_hat / mu_hat          # coefficient of variation

The coefficient of variation (CV) is particularly useful for comparing dispersion across responses with different units. A CV of 0.15 on HIC₁₅ indicates moderate scatter; a CV above 0.30 usually signals that a dominant input is poorly characterized and should be prioritized in sensitivity analysis. Note the (N−1) denominator: it yields an unbiased variance estimate, which matters when N is small (tens of runs, typical for FE-based UQ).

Correlation and dependence

Occupant injury responses are rarely independent across outputs. Chest deflection and rib fracture count are driven by overlapping physics; peak head acceleration and HIC₁₅ are functionally linked. The covariance matrix Σ captures these relationships, and its off-diagonal entries encode the linear dependence structure. When inputs are correlated, sampling must respect Σ — naive independent sampling of a multivariate normal with the wrong covariance will understate or overstate response variance. The Pearson correlation coefficient ρ = Cov(X,Y)/(σ_Xσ_Y) is the normalized version; values near ±1 indicate strong linear dependence, while ρ near 0 does not preclude nonlinear dependence.

What this foundation enables

With random variables defined, moments understood, and distribution families selected on physical grounds, the UQ workflow gains its basic currency: a response is a distribution, a threshold comparison is a CDF evaluation, and a risk estimate is an integral of a risk curve against a response density. The next steps — sampling design, surrogate modelling, and sensitivity analysis — all operate on exactly these objects. Getting the distributional assumptions right at this stage prevents the more expensive errors that surface later when tail probabilities are extrapolated from a mis-specified model.