Normal random variable

Table 20.5.2 also can be used to determine probabilities concerning normal random variables tliat are not standard normal variables. The required probability is first converted to tm equivalent probability about a standard normal variable. For example if T, the time to failure, is normally distributed with mean p = 100 and stanchird deviation a = 2 tlien (T - 100)/2 is a standard normal variable and... 

Tlie probabilities given in Eqs. (20.5.10), (20.5.11), and (20.5.12) are tlie source of the percentages cited in statements 1, 2, and 3 at tlie end of Section 19.10. These can be used to interpret tlie standard deviation S of a sample of observations on a normal random variable, as a measure of dispersion about tlie... 

Note tliat as previously mentioned, increasing tlie number of simulated values increases tlie accuracy of the estimate. Also note tliat Monte Carlo simulation provides an attractive alternative to solving tlie somewhat complicated matliematical problem of finding tlie expected value of the minimuin of two normal random variables. 

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. 

The physical and conceptual importance of the normal distribution rests on one unique property the sum of n random variables distributed with almost any arbitrary distribution tends to be distributed as a normal variable when n- oo (the Central Limit Theorem). Most processes that result from the addition of numerous elementary processes therefore can be adequately parameterized with normal random variables. On any sort of axis that extends from — oo to + oo, or when density on the negative side is negligible, most physical or chemical random variables can be represented to a good approximation by a normal density function. The normal distribution can be viewed a position distribution. 

The ratio of two normal random variables with zero mean is distributed as a Cauchy variable. Isotopic ratios such as 206Pb/204Pb and 207Pb/204Pb therefore should not be described as normal variables since ratios of ratios (e.g., 207Pb/206Pb) should be distributed with a consistent distribution. A consistent distribution for isotopic ratios is the log-normal distribution. 

If X is a normal random variable with zero mean and unit variance, find the distribution fY(y) of the variable Y related to X through the function... 

X is a normal random variable with mean p and variance a2. Given the set of samples of m observations with mean X and variance S2, which will be treated as independent random variables, show that the ratio... 

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). 

Transformations of the data may be used to extend the applicability of a particular standard distribution, in practice usually the normal distribution. For example, a log-normal random variable is a random variable that is normal after logarithmic transformation. Power transformations are also widely used, e.g., with Box-Cox transformations. 

Normal Random Variable. The probability density function of a normally distributed random variable, y, is completely characterized by its arithmetic mean, y, and its standard deviation, a. This is abbreviated as N (y,cr2) and written as ... 

Suppose V n = 0, — 1, -2,. .. is a set of real, normalized random variables subject to a stationary distribution with covariance... 

Here p and are, respectively, the mean value and the dispersion (variance) with respect to a population. These characteristics establish all the integral properties of the normal random variable that is represented in our example by the value expected for the species concentration in identical samples. It is not feasible to calculate the exact values of p and because it is impossible to analyse the population of an infinite volume according to a single property. It is important to say that p and show physical dimensions, which are determined by the physical dimension of the random variable associated to the population. The dimension of a normal distribution is frequently transposed to a dimensionless state by using a new random variable. In this case, the current value is given by relation (5.21). Relations (5.22) and (5.23) represent the distribution and repartition of this dimensionless random variable. Relation (5.22) shows that this new variable takes the numerical value of x when the mean value and the dispersion are, respectively, p = 0 and = 1. 

The above relation shows that each of the n divisions of the population has the CT /n dispersion. Now, considering that a division x — p is a normal random variable and that the mean value of this variable is zero, we can transform relation (5.22) into relation (5.37) where u keeps its initial properties (mean value is zero and dispersion equal to unity) ... 

At this step of the data preparation, we can observe that each column of the transformed statistical data has a zero mean value and a dispersion equal to one. A proof of these properties has already been given in Section 5.2 concerning a case of normal random variable normalization. 

The probability is obtained by integrating the distribution function over the appropriate limits. For example, for x being a normal random variable with mean px and standard deviation (Tx, the probability that x > a cein be expressed as... 

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. 

Transform the Nt sample vectors x k =, 2,...,Nf) defined in the original (i.e., physical) space of possibly dependent, non-normal random variables (step 2. above) into Nf samples 0 k= 1,2,..., iVr defined in the standard normal space where each component o f the vector =... 

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... 

The probability distributions of concrete column resistance Rn or Rm and permanent action effect Nq or Mg are close to a normal distribution (Ellingwood, 1981 ISO 2394 1998 JCSS2000). The distribution of the safety margins is close to lognormal but they may be treated as a normal random variables (Swed et al. 2005 Sugiyama Yoshida 2008). 

Note that we can use standard normal loss function because demand over the lead time (DLT) is the convolution of normal random variables for demand and lead time, and is therefore normally distributed. See the Further Readings section of this chapter for references that address situations where lead-time demand is not normal. 

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, standard normal random variable. For instance, the left panel in O Fig. 9.1 makes use of standardization to show the properties of one-dimensional 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.)... 

---