1. Calculation of gPC Coefficients

The gPC expansion is given by . The coefficients are found using the projection formula:

The problem states the Legendre polynomials are normalized, which means their expected square is one: . Therefore, the formula simplifies to:

We will use the general rule for expectations of powers of : for even , and for odd .

Using linearity of expectation:

This can also be written as .

Since both terms are odd powers, their expectations are zero:


2. Statistics from gPC Coefficients

With the coefficients known, we can easily calculate the statistics of the model output .

Mean Value

The mean is given by the first coefficient, .

Variance

The variance is the sum of the squares of all higher-order coefficients ().

Raw Second Moment

The raw second moment is the sum of the squares of all coefficients ().

This result is consistent with the direct calculation of .