By Robert L. Zimmerman, Fredrick I. Olness

A suitable complement for any undergraduate and graduate path in physics, ** Mathematica® for Physics** makes use of the ability of

**to imagine and exhibit physics strategies and generate numerical and graphical recommendations to physics difficulties. through the ebook, the complexity of either physics and**

*Mathematica®***is systematically prolonged to develop the diversity of difficulties that may be solved.**

*Mathematica®*

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Order the elements of U ∗ without repeats and relabel them using the notation u∗i so that u∗1 < u∗2 < · · · < u∗l+1 , where l = U ∗ − 1, and · denotes cardinality. • Let Ik = {(i, j) | uij ≤ u∗k and u∗k+1 ≤ ui(j+1) } for k = 1, 2, . . , l. Then for u ∈ (u∗k , u∗k+1 ), the PDF of U is given by fU (u) = fUij (u) (i,j)∈Ik for k = 1, 2, . . , l. 2 Data Structure In order to implement the algorithm, we will use a data structure for the distribution of the bivariate random variable that expands on the list-of-sublists format used in APPL and described in Chap.

2. If ti = −∞ and Ti is ﬁnite, then xi = (Ti + Tˆi )/2 and yi is the average of the two y-values that correspond to xi on the two constraint equalities associated with xi . 3. If ti is ﬁnite, then xi = (ti + tˆi )/2 and yi is the average of the two y-values that correspond to xi on the two constraint equalities associated with xi . Each constraint deﬁning Ai is an inequality of the form p(x, y) < 0, where p is a real-valued continuous function. The corresponding constraint for Bi is found by substituting the appropriate inverse transformations determined by the algorithm described in the previous paragraph to achieve the inequality p ri (u, v), si (u, v) < 0.

52, pages 128–129] extend this many–to–1 technique to n-dimensional random variables. We are concerned with a more general univariate case in which the transformations are “piecewise many–to–1,” where “many” may vary based on the subinterval of the support of Y under consideration. We state and prove a theorem for this case and present APPL code to implement the result. Although our theorem is a straightforward generalization of Casella and Berger’s theorem, there are a number of details that have to be addressed in order to produce an algorithm for ﬁnding the PDF of Y .