increases quickly with the number of input random variables
M and the degree l of the polynomials; For example, M = 10 and l = 5 lead to L = 3003.
[mu] and [sigma] are the mean and standard deviation of the normal random variable
. N is the quadrature order (equal to the number of Gauss point).
Let [X.sub.1] ~ g(x) and [X.sub.2] ~ (1/[kappa])h(x)g(x) be independent random variables
Comparing random variables
is the essence of the actuarial profession; in order to acquire more general results, we give the notion of weak correlation orders as follows.
The polynomial chaos methods are well suited for the random differential equations, RDE, with a very few number of random variables
defining their main coefficients.
For a [member of] (1/3,1/2] and n [member of] N define a random variable
[Z.sup.([alpha])].sub.n] : [[0,1].sup.N] [right arrow] R by the formula:
If t [member of] T is fixed, then [X.sub.t] is a random variable
, whereas if we fix e [member of] S, [X.sub.e] is a function of time, called a realization or sample path of the process X(t, e).
The first four moments of random variable
Z (mean, standard deviation, skewness and kurtosis) can be expressed in terms of the expansion coefficients [b.sub.i], i = 0,1,2,....
FSD is a well-accepted way to define when one random variable
is larger than another.
The assumption of random variables
and random disturbance is vital for valid model estimation and testing, and thus this assumption is crucial for macroeconometrics.
where [f.sub.X](X) denotes the joint PDF of the random variables
Since round-off errors for random variables
that are rounded to the nearest integer are distributed Uniform (-1/2, 1/2), the sum of round-off errors is a linearly transformed Irwin-Hall distribution .