Once the PCEs for the model inputs are determined, the next step is to
approximate the probability density distribution of the model response variable. In this
case, we use the problem-specific polynomials derived from the probability density functions
of the uncertain input parameters. For example, if the input parameters were characterized
by normal distributions, then we use Hermite polynomials for the model output expansions, as
determined by Table 20.1: Summary of General Orthogonal Expansions
. The general form of the expansion
for an output variable 
is then
 | (20–11) |
where 
are the standard, independent, random variables used to describe the
variant model inputs and 
are coefficients that need to be determined. In general, Equation 20–11
will include terms
corresponding to each of the input parameters plus cross-link terms for all the combinations
that are relevant for the specified order of approximation. For example, a 3rd-order Hermite
polynomial expansion of a model response that considers two random input parameters would
have terms corresponding to those listed in Table 20.2: Terms for a 3rd-Order Hermite Polynomial Expansion with Two Parameters
below.
