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 .