Algebra of random variables

Theoretically, the effects of the manufacturing process on the material property distribution can be determined, shown here for the case when Normal distribution applies. For an additive case of a residual stress, it follows that from the algebra of random variables (Carter, 1997) ... 

Figure 4.28 shows the derivation of equation 4.38 from the algebra of random variables. (Note, this is exaetly the same approaeh deseribed in Appendix VIII to find the probability of interferenee of two-dimensional variables.)... 

Using the algebra of random variables we ean solve the probability of interferenee between the two toleranee distributions, assuming that the variables follow a... 

In this section, the algebra of random variables will be rehearsed from the viewpoint of error propagation. Convolution, central limit theorem, as well as random sums (i.e., sums of a random number of random variables) are also included here because of their importance in nuclear applications. 

Virtually all-classical design equations assume single-valued, real numbers. Such numbers can be multiplied, divided, or otlierwise subjected to real-number operations to yield a single-valued, real number solution. However, statistical materials selection, because it deals witli the statistical nature of property values, relies on tlie algebra of random variables. Property values described by random variables will have a mean value, representing the most typical value, and a standard deviation tliat represents the distribution of values around the mean value. 

This requires treating the mean values and standard deviations of particular property measurements according to a special set of laws for the algebra of random variables. Extensive information can be found in... 

We need this speeial algebra to operate on the engineering equations as part of probabilistie design, for example the bending stress equation, beeause the parameters are random variables of a distributional nature rather than unique values. When these random variables are mathematieally manipulated, the result of the operation is another random variable. The algebra has been almost entirely developed with the applieation of the Normal distribution, beeause numerous funetions of random variables are normally distributed or are approximately normally distributed in engineering (Haugen, 1980). 

It is quite common to see, in the literature, the results of a regression analysis being discussed in terms of the correlation of the dependent variable with the independent variable. Strictly, this does not make sense, because correlation is defined for pairs of random variables, whereas in regression only the dependent variable is supposed to be random. However, if we set aside this conceptual detail, some algebraic relations between correlation and regression do exist that are worthwhile discussing, if only to clarify their true meanings and hmitations. 

The theories based on the equation of state are more versatile. The model developed by Simha and many of his collaborators is most useful. By contrast with the H-F theory it leads to two binary interaction parameters, one energetic the other volumetric, that are constant in the full range of independent variables. Furthermore, it has been found that the numerical values of these two parameters can be approximated by the geometric and algebraic averages, respectively. The non-random mixing can easily be incorporated into the theory. 

The probability space (Q, E, P) is composed of a set of elementary events Q, a cr-algebra E and a probability measure P (Paimier et al. 2013). The sample space Q contains all possible elementary events continuous random variables, e.g., Q M, the Borel ff-algebra, denoted by S(R), contains all possible intervals / CM. The proba-bUity measure P assigns for each event s E a. real value in [0,1], representing the probability of s. The probability measure is the mapping... 

Once the algebraic problem is solved and the approximation coefficients have been determined, an expression of the displacement field that depends on the input random variables has been obtained. This expression can be considered as a response surface. This response surface has local character (Proppe 2008) and depends on the size and location of the elements y, if a partition of F is adopted. 

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