Standard normal random variable

The eigenvalue/eigenvector decomposition of the covariance matrix thus allows us to redefine the problem in terms of Nc independent, standard normal random variables 0in. 

When k = 2, the Kruskal-Wallis chi-square value has 1 df This test is identical to the normal approximation used for the Wilcoxon Rank-Sum Test. As noted in previous sections, a chi square with 1 df can be represented by the square of a standardized normal random variable. In the case oik = 2, the //-statistic is the square of the Wilcoxon Rank-Sum Z-test (without the continuity correction). 

Considerable work has been focused on determining the asymptotic null distribution of -2 log-likelihood -ILL) when the alternative hypothesis is the presence of two subpopulations. In the case of two univariate densities mixed in an unknown proportion, the distribution of -ILL has been shown to be the same as the distribution of [max(0, Y)f, where Y is a standard normal random variable (28). Work with stochastic simulations resulted in the proposal that -2LL-c is distributed with d degrees of freedom, where d is equal to two times the difference in the number of parameters between the nonmixture and mixture model (not including parameters used for the probability models) and c=(n-l-p- gl2)ln (31). In the expression for c, n is the number of observations, p is the dimensionality of the observation, and g is the number of subpopulations. So for the case of univariate observations (p = 1), two subpopulations (g = 2), and one parameter distinguishing the mixture submodels (not including the mixing parameter), -2LL-(n - 3)/n with two... 

For a standard normal random variable z) is the area under the Standard Normal Curve from —oo to z). 

This is called the chi-square distribution with v degrees of freedom and is important for statistical hypothesis testing. It can be shown that if Z is a standard normal random variable, then is a random variable having the chi-square distribution with one degree of freedom. Further, the exponential distribution with /3 = is the chi-square distribution with two degrees of freedom. 

Example 2.1 Suppose that 911 calls arrive in a Poisson process of rate one per minute. The probability of more than 80 calls in a 60-minute period may be calculated approximately as follows. Let Z denote a standard normal random variable. By (12) and (18), the mean and vruiance of 1V(60) are both approximately equal to 60, and so we have P(N(60) > 80) P(Z > (80 — 60)/V ) = P(Z > 2.582) = 0.005. Of course, in this case we are dealing with a special renewal process, namely the Poisson process. Therefore, property 1 of Subsection 2.1 may be used as an riltemative approach to computing this probability. 

Next, Ui and are transformed into standard normal random variables Z, and W and then the joint distribution of Z and W introduces the correlation through a bivariate normal distribution. 

Solve the conditional one-dimensional reliability problem associated to each random sample, k = 1,2,...,Nr, in which the only (standard normal) random variable is c. The associated conditional failure probability P (F), k = 1,2,..., Nt, is given by... 

Standardization usually leads out from the distribution family of X. For instance, if X is distributed as Poisson, then 7 is certainly not. However, if X happens to be N(/i, normal distributions. As standardization shifts the distribution such that the mean becomes 0, this transformation applied to the exponential distribution on the right panel in the same figure would result in a shifted exponential, which however is not considered an exponential distribution. (The exponential distribution is kind of attached to the soil as its mode is at f = 0 by definition.)... 

Standardization. Shifting and/or rescaling a normal random variable will result in another normally distributed random variable. Therefore, the standardization of the N(fi, c normal random variable X will result in an N 0,1) standard normal random variable Y ... 

A useful byproduct of the FORM analysis is the normal vector of the linear approximation of the LSF a (Der Kiureghian, 2005). The elements of this vector can be interpreted as importance measures of the standard normal random variables U- (Figure 1) ... 

The performance function expressed by equation (11) or (14), enables an evaluation of the dynamic structural reliability that includes parameter uncertainties can be performed using FORM. Calculation of the gradients of the performance function is an important step of the FORM. However, it is not always easy to obtain these gradients, (especially for the case where non-linearity of the structural performance is considered). Yao and Wen (1996) have introduced response surface approach (RSA) to avoid the sensitivity analysis required in FORM, where the response surface function is expressed as a polynomial of basic random variables in original space. For simplification, the performance function shown in equation (11) is approximated by the following second-order polynomial of standard normal random variables, in which the mixed terms are neglected. 

Since Pfi is a function of only one standard normal random variable ui, its mean value can be point-estimated from Eq. 35. For a problem with n variables, if the probability moments of is estimated using w-point estimate, only mn function calls of P/x) are required for estimating the general probability of failure. 

Example 5.3. Consider the reliabihty assessment problems from (Sorensen 2004, Note 6) (series system) and (Sorensen 2004, Note 7) (parallel system), with X = (X,X2), where Xj and Xj are independent standard normal random variables defined in 2 = R, and the limit state functions of Table 3. 

To simplify the formulation, the basic random variables (0, v) are transformed into space of standard normal random variables denoted by U= (l/ , Uv) where Uo and are the vectors of standard normal random variables originating from 0 and v, respectively, with the understanding that the transformations 0 = T0 Us) and v = Tj, Uv) exist (Melchers 1999). The performance function in terms of the standard normal variable vector f/is denoted by G U) = h — mai A where h U0,Xk) = h[T0 U0),xi,, = 0,1, -,N. Instead of drawing samples from pi/u), one could also select an importance sampling PDF tij/Cm) and evaluate Pp using... 

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