Univariate distributions

As described in earlier chapters, for univariate QMOM the closure problem appears in different parts of the transport equations for the moments of the NDF. However, it can often be reduced to the following integral  

In Gaussian quadrature theory the NDF is called the weight function or measure. The weight function must be nonnegative and non-null in the integration interval and all its moments. 

The integration domain and the weight function n( ) uniquely define the family of polynomials PaiO)- A polynomial is defined as monic when its leading coefficient (i.e. kap) is equal to unity. Below two important theorems (without proof) are reported. 

Theorem 3.1 Any set of orthogonal polynomials Paif) has a recurrence formula relating any three consecutive polynomials in the following sequence  

The recursive relation is the most important property for constructive and computational use of orthogonal polynomials. In fact, as will be shown below, knowledge of the recursion coefficients allows the zeros of the orthogonal polynomials to be computed, and with them the quadrature rule. Therefore the calculation of the coefficients of this three-term recurrence relation is of paramount importance. The recursive relationship in Eq. (3.5) generates a sequence of monic polynomials that are orthogonal with respect to the weight function 

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N.L. Johnson and S. Kotz, Distributions in Statistics Continuous Univariate Distributions — I, Wiley, New York, 1970. 

SOME PRACTICAL ASPECTS OF TH E SELECTION OF UNIVARIATE DISTRIBUTIONS... 

The joint distribution for two inde]x ndeut Gaussian variables is just t he [rrod-uct of two univariate distributions. When the clata forming a vcctoi d art correlated... 

The variable x in the preceding formulas denotes a quantity that varies. In our context, it signifies a reference value. If the variable by chance may take any one of a specified set of values, we use the term variate (i.e, a random variable). In this section, we consider distributions of single variates (i.e., univariate distributions). In a later section, we also discuss the joint distribution of two or more variates bivariate or multivariate distributions). 

Transformation to approximately Gaussian shape of the univariate distributions was necessary. 

Figure 2 The data from Table I as a scatter plot and, along each axis, the univariate distributions. Two distoKt groups are evident from the data...

The theory of Gaussian quadrature applies only to univariate distributions. However, in practical cases, the study of distributions with multiple internal coordinates is often necessary. In these cases the closure problem generally assumes the following form ... 

Continuous Univariate Distributions-2 Houghton Mifflin Co. Boston, 1970 p. 44. 

Examination of the univariate distribution of 5-FU clearance revealed it to be skewed and not normally distributed suggesting that any regression analysis based on least squares will be plagued by non-normally distributed residuals. Hence, Ln-transformed 5-FU clearance was used as the dependent variable in the analyses. Prior to analysis, age was standardized to 60 years old, BSA was standardized to 1.83 m2, and dose was standardized to 1000 mg. A p-value less than 0.05 was considered to be statistically significant. The results from the simple linear regressions of the data (Table 2.4) revealed that sex, 5-FU dose, and presence or absence of MTX were statistically significant. 

C-4] Abramowitz M, Stegun lA (eds) (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Department of Commerce, Washington [C-5] Johnson NL, Kotz S, Balakrishnan N (1995) Continuous univariate distributions, vol 2. Wiley, New York... 

Johnson, N.L. and Kotz, S. (1970). Distributions in Statistics. Continuous Univariate Distributions - I. John Wiley Sons, Ltd, New York. 

Fundamentals of continuous thermodynamics as applied to homopolymers characterized by univariate distribution functions have been reviewed extensively [28, 29]. Hence, this chapter will provide the fundamentals in their most general form by considering systems composed of any number of polydisperse ensembles described by multivariate distribution functions and any number of solvents and by referring to the papers [28, 29],... 

The expressions for InpR associated with the equation-of-state models and the univariate distribution function are given in the papers [56-59]. 

More generally, if 6r( ) is the cumulative distribution function of any univariate distribution, and if 0 ) denotes the inverse of that function, then the numbers... 

Jhaveri, B.S., and G.K. Youngren Three-Parameter Modification of the Peng-Robin-son Equation of State to Improve Volumetric Predictions, SPERes. Eng.,-p. 1033,1988. 23. Johnson, N.L., and S. Kotz Continuous Univariate Distributions, Houghton Mifflin Co., Boston, 1970. 

The usage of univariate distributions for sampling initial conditions is based on the assumption that the uncertain parameters are not correlated with each other. Quite often uncertain parameters are subject to correlations and thus the univariate approach is not applicable. This happens when a generic outcome is dependent on different phenomena simultaneously (i.e. the outcome dependency description can not be collapsed to a function of a single variable). RAVEN currently supports both N-Dimensional (N-D) PDFs. The user can provide the distribution values on either Cartesian or sparse grid, which determines the interpolation algorithm used in the evaluation of the imported CDF/PDF ... 

As already mentioned, the sampling methods use the distributions in order to perform probability-weighted perturbations. For example, in the Monte Carlo approach, a random number e [0,1] is generated (probabihty threshold) and the CDF, corresponding to that probabihty, is inverted in order to retrieve the parameter value usable in the simulation. The existence of the inverse for univariate distributions is guaranteed by the monotonicity of the CDF. For N-D distributions this condition is not sufficient since the CDF(X) —> [0,1 ], Xe R and therefore it could not be a bijective function. From an application point of view, this means the inverse of a N-D CDF is not unique. 

Johnson, N.L. Kotz, S., Balakrishnan, N. Continuous 1994. Univariate Distributions Vol. 1, 2nd Edition, J. Wiley, New York, Chichester, Brisbane, Toronto, Singapore. 

Results of uni- and bi-variate distribution analyses are often used as proof of causal relationships. Let us look at two examples. We have earlier quoted a study that shows that 88 per cent of the accidents are caused primarily by unsafe acts (Heinrich, 1959). This statement is based on an uni-variate analysis of accidents by immediate cause. In the example in Section 15.2, an univariate distribution analysis showed that inattention was the cause of almost a quarter of the accidents. These examples bring us to the basic questions about what we mean by an accident cause. 

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