MULTI-ELEMENT GENERALIZED POLYNOMIAL CHAOS FOR ...

914 XIAOLIANG WAN AND GEORGE EM KARNIADAKIS
3.1.3. Orthogonal polynomials for Gaussian distribution. Some random
distributions, e.g., Gaussian distribution and Gamma distribution, have long tails.
Next, we demonstrate the decomposition of random space and the corresponding
orthogonal polynomials for this type of random variable. Given a Gaussian random
variable X ∼ N(0, 1), we decompose the random space into three random elements
(−∞, −a], [−a, a], and [a, +∞), where a is a positive constant. In the middle element
[−a, a], we define a random variable Xm with a conditional PDF
(3.8)
fm(xm) =
1 − e 2 x2
m
� a 1
e− 2
−a x2dx
.
In one of the tail elements, say, [a, +∞), we define a random variable X∞ with a
conditional PDF
(3.9)
f∞(x∞) =
1 − e 2 x2
∞
� ∞ 1
e− 2
a x2dx
.
We choose a in such a way that Pr(X ≥ a)