For

normal random variables, X can be transformed into standard

normal random variables u through a linear transformation, as follows:

A normal or non-normal random variable Z can be approximated using one-dimensional Hermite orthogonal polynomials [[PSI].sub.i] of the standard

normal random variables [xi] as [17,18]

where [[alpha].sub.j] = [m.sub.j]/(1 + [b.sup.2.sub.j]), [[beta].sup.2.sub.j] = 2[[psi].sup.-1.sub.j] - 1 + 2[square root of [[psi].sup.-1.sub.j]] [square root of 2[[psi].sup.-1.sub.j] - 1], and [Z.sub.Vj] are independent standard

normal random variables.

In order to measure the (in)efficiency of Parliament in Asian countries, in this paper we propose a stochastic frontier production function for unbalanced panel data [5] assumed to be distributed as a half

normal random variables. This type of frontier and the computation method present advantages with respect other alternatives, for example the deterministic frontiers [12,16].

zero-mean

normal random variables x , y , and z with standard deviation [[sigma].sub.xyz] so that the new position of the receiver is now at coordinates (D + x',y',z').

For all scenarios, we assume that the bivariate

normal random variables (X, Y) have population parameters [[mu].sub.x] = 0, [[sigma].sub.x.sup.2] = [[sigma].sub.y.sup.2] = 1.

are

normal random variables with mean zero and covariance structure :

To view this issue from a different perspective, a chi-squared random variable with n degrees of freedom arises from summing the squares of n independent

normal random variables. Hence, a chi-squared random variable with 1 degree of freedom is simply the square of a

normal random variable.

where Z = [([Z.sub.1], ..., [Z.sub.m]).sup.T] is a random vector consisting of m mutually independent standard

Normal random variables, A is an n x m matrix, [mu] is an n x 1 vector and [??] stands for "equality in distribution." The relationship between [SIGMA] and A is given by [SIGMA] = [AA.sup.T].

Moreover, if the relative price changes in each holding period are independently and identically distributed

normal random variables, the empirical volatilities, [S.sub.23], are related to the chi-square random variable [[chi].sub.k.sup.2] = by [[chi].sub.k.sup.2] = k h [S.sub.23.sup.2] where k = n -1 and n is the number of trading days in the holding period, in this case 23.

randn for

normal random variables, and rand for continuous uniform