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AbstractFor a young science and even for most old ones, the intensity and variety of activities involved in computer science are unsurpassed. In an effort to provide the extremely varied training needed in the field, curriculum planners have tended to slight probability and statistics. Rarely has it been a requisite and only occasionally a desired electiveHowever, not only is an adequate foundation in probability and statistics necessary for such external computer applications as tracking and other data reduction, Monte Carlo techniques, gaming, operations research, traffic analyses, etc., but it also plays an important role in internal applications to computer science and technologyThis effort in Rand's continuing study of computational mathematics and its applications offers a number of examples in various areas of computational mathematics and computing machine arithmetic‐to say nothing of computer design, time sharing, and the like‐to support the thesis that probability and statistics should be a requisite, not simply an elective or, even worse, ignored, at any institution offering a program in computer science. Further more, a potentially profitable source of research problems can be found in the interface between probability and statistics and computer science.

I. Elementary mathematics -- 1. Fundamental operations on rational numbers -- 2. Higher arithmetical operations -- 3. Development of the number system -- 4. Algebraic equations -- 5. Functions -- 6. Percentages, interest and annuities -- 7. Plane geometry -- 8. Solid geometry -- 9. Descriptive geometry -- 10. Trigonometry -- 11. Plane trigonometry -- 12. Spherical trigonometry -- 13. Analytic geometry of the plane -- II. Steps towards higher mathematics -- 14. Set theory -- 15. The elements of mathematical logic -- 16. Groups and fields -- 17. Linear algebra -- 18. Sequences, series, limits -- 19. Differential calculus -- 20. Integral calculus -- 21. Series of functions -- 22. Ordinary differential equations -- 23. Complex analysis -- 24. Analytic geometry of space -- 25. Projective geometry -- 26. Differential geometry, convex bodies, integral geometry -- 27. Probability theory and statistics -- 28. Calculus of errors, adjustment of data, approximation theory -- 29. Numerical analysis -- 30. Mathematical optimization -- III. Brief reports on selected topics -- 31. Number theory -- 32. Algebraic geometry -- 33. Further algebraic structures -- 34. Topology -- 35. Measure theory -- 36. Graph theory -- 37. Potential theory and partial differential equations -- 38. Calculus of variations -- 39. Integral equations -- 40. Functional analysis -- 41. Foundation of geometry — Euclidean and non-Euclidean geometry -- 42. Foundations of mathematics -- Tables.

one Tables and Graphs -- I. Tables -- II. Graphs -- two Elementary Mathematics -- I. Approximate computations -- II. Algebra -- III. Geometry -- IV. Trigonometry -- three Analytic and Differential Geometry -- I. Analytic geometry -- II. Differential geometry -- four Foundations of Mathematical Analysis -- I. Introduction to analysis -- II. Differential calculus -- III. Integral calculus -- IV. Differential equations -- five Supplementary Chapters on Analysis -- I. Complex numbers and functions of a complex variable -- II. Vector calculus -- III. The calculus of variations -- IV. Integral equations -- V. Fourier series -- six Interpretation of Experimental Results -- I. Foundations of the theory of probability and the theory of errors -- II. Empirical formulas and interpolation.

Summary This is an attempt to write an introduction to some aspects of the present state of the Neyman‐Pearson theory. The object is to interest the reader who has some knowledge of modem probability theory and statistics. We try to emphasize the main ideas, "unnecessary" mathematics is avoided.In our opinion, the Neyman‐Pearson theory (or, more generally, the objectivistic approach to statistics) is a vigorous attempt to statisfy some weakened form of the philosophical principle of the "intersubjective verifiability". This attempt is considered successful in those situations where "various different objectivistic optimum properties are satisfied by the same optimal test". Unfortunately, these conditions are not satisfied for many situations from actual practice (testing against restricted alternatives, non‐parametric problems). Thus we arrive at a dilemma: for many problems from practice different "optimum" tests are available. There is no easy way out of this dillemma (see the last section).We have tried to emphasize the motivation and to aim at readability while minimizing the overlap with e.g. Lehmann's basic textbook. Thus many basic tools are disregarded: exponential families, completeness and sufficiency of statistics and many other subjects are missing, the restrictions by similarity, unbiasedness or invariance are only mentioned. The references constitute an incomplete apology for deleting so many important contributions.