The Philosophy of Probability Values Behaviour, through Fractions and Composite Probability Function in the Continuous Case
Keywords:
Geometric Probabilities, Negative Probabilities, Dependent
Abstract
This paper deals with probability theory, and it is an extension to a published paper that has the same title, but for the discrete case. This present paper is aiming to study probability values behavior, in the case of continuous sample space, through fractions intervals and composite function. This aim tends to study the value behavior rather than finding the value itself. Also, this aim requires usage of some concepts of continuity, geometric probability, and measure theory, which also need a brief treatment. This paper is mainly using an experiment with a design that helps to study the probability fractions values in the form of intervals in the case of one direction movement and in the case of different directions. As a result, every case reflects some aspects of probability values behavior and can clarify many important characteristics of the probability theory. In addition to applying the composite function by some important theorems of conditional probability. These are besides a proven proposition that helps to design experiment, upon the understanding of the case nature. In addition to a corollary that allows to visualize negative probability values, as a particular case (trial), that upon the validity of the explanation of negativity which should be consistent with the probability axioms.
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References
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25. Keller, J. B. (1986). The Probability of Heads. The American Mathematical Monthly, 93(3), 191–197. https://doi.org/10.2307/2323340
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31. Lebesgue, Henri. (1989). Leçons sur L’intégration et La Recherche des Fonctions Primitives. Deuxième Édition. Paris: Éditions Jacques Gabay.
32. Lévy; Paul. (2004,2006). Calcul Des Probabilités. Paris: Éditions Jacques Gabay.
33. Loève, M. (1977). Probability theory, Volume I. 4th Edition. New York: Springer-Verlag.
34. Loève, M. (1978). Probability theory, Volume II. 4th Edition. New York: Springer-Verlag.
35. Mcginty, M. (2004). Geometric Probability for The Space-Time Plane. Pi Mu Epsilon Journal, 12(1), 25–35. http://www.jstor.org/stable/24340793
36. Meyer, P. L. (1970). Introductory Probability and StatisticalApplications. Second Edition. USA: Addison-Wesley Publishing Company, Inc.
37. Monticino, M. (2001). How to Construct a Random Probability Measure. International Statistical Review / Revue Internationale de Statistique, 69(1), 153–167. https://doi.org/10.2307/1403534
38. Parzen, E. (1960). Modern Probability Theory and Its Applications. New York, NY: John Wiley & Sons, Inc.
39. Parzen, E. (1962). Stochastic Processes. USA: Holden-Day, Inc.
40. Rényi, Alfréd. (1970). Probability Theory. Amsterdam: Holden-Day and Akadémiai Kiadó.
41. Ross, Sheldon M. (1989). A First Course in Probability. Third Edition. New York: Macmillan Publishing Company.
42. Shiryaev, Albert N. (2016). Probability-1. Third Edition. New York: Springer.
43. Stoll, Manfred. (1997). Introduction to Real Analysis. Boston: Addison Wesley Longman Inc.
44. Sveshnikov, A. A. (Ed.). (1968). Problems in Probability Theory, Mathematical Statistics and Theory of Random Functions. USA: W. B. Saunders Company.
45. Swokowski, Earl W. (1988). Calculus with Analytic Geometry. Second Alternate Edition. USA: PWS-KENT Publishing Company.
46. Szekely, Gabor. (2005). Half of a coin: Negative probabilities. Wilmott Magazine. 50. 66-68.
47. Tucker, Howard G. (1967). A Graduate Course in Probability. New York: Academic Press, Inc.
48. Uspensky, J. V. (1937). Introductions to Mathematical Probability. First Edition, Second Impression. New York: McGraw-Hill Book Company,Inc.
49. Velleman, D.J. (1997). Characterizing Continuity. Amer, Math, Monthly 104, 318-322.
2. Bartle, Robert G., & Sherbert, Donald R. (2011). Introduction to Real Analysis. Fourth Edition. USA: John Wiley & Sons, Inc.
3. Baten, W. D. (1930). Simultaneous Treatment of Discrete and Continuous Probability by Use of Stieltjes Integrals. The Annals of Mathematical Statistics, 1(1), 95–100. http://www.jstor.org/stable/2957676
4. Bingham, N. H. (2000). Studies in the History of Probability and Statistics XLVI. Measure into Probability: From Lebesgue to Kolmogorov. Biometrika, 87(1), 145–156. http://www.jstor.org/stable/2673568
5. Blass, Andreas & Gurevich, Yuri. (2015). Negative probability. Bull Eur Assoc Theor Comput Sci. 115.
6. Block, H. W., Savits, T. H., & Shaked, M. (1982). Some Concepts of Negative Dependence. The Annals of Probability, 10(3), 765–772. http://www.jstor.org/stable/2243384
7. Bohr, N. (1958). Atomic Physics and Human Knowledge. New York: John Wiley and Sons.
8. Borel, Émile. (2003, 2006). Leçons sur La Théorie des Fonctions. Quatrième Édition. Paris: Éditions Jacques Gabay.
9. Botts, T. (1969). Probability Theory and the Lebesgue Integral. Mathematics Magazine, 42(3), 105–111. https://doi.org/10.2307/2689118
10. Cauch, A. L. (1821). Cours d’Analyse. Paris: Gauthier-Villars.
11. Cramér, H. (1946). Mathematical Methods of Stgatistics. First Printing. USA: Princeton University Press.
12. Davenport, Wilbur B., Root, William L. (1958). An Introduction to the Thwory of Random Signals and Noise. USA: McGraw Hill Book Company.
13. Dirac, P. A. M. (1942). Bakerian Lecture. The Physical Interpretation of Quantum Mechanics. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 180(980), 1–40. http://www.jstor.org/stable/97777
14. Eagle, A. (2005). Randomness Is Unpredictability. The British Journal for the Philosophy of Science, 56(4), 749–790. http://www.jstor.org/stable/3541866
15. Edwards, William. F., Shiflett, Ray. C., & Shultz, Harris. S. (2008). Dependent Probability Spaces. The College Mathematics Journal, 39:3, 221-226, Doi: 10.1080/07468342.2008.11922296.
16. Feller, W. (1950). An Introduction to Probability Theory and Its Applications, Volume I. New York, NY: John Wiley & Sons, Inc.
17. Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Volume II. Second Edition. USA: John Wiley & Sons, Inc.
18. Feynman, Richard P. (1987). Negative Probability. In Peat, F. David; Hiley, Basil (eds.). Quantum Implications: Essays in Honour of David Bohm. Routledge & Kegan Paul Ltd. pp. 235–248.
19. Gnedenko, B. V. (1963). The Theory of Probability. Second Edition. USA: Chelsea Publishing Company.
20. Gnedenko, B. V., & Kolmogorov, A. N. (1968). Limit Distributions for Sums of Independent Random Variables. USA: Addison-Wesley Publishing Company.
21. Goffman, C. (1953). Definition of the Lebesgue Integral. The American Mathematical Monthly, 60(4), 251–252. https://doi.org/10.2307/2307435
22. Jughaiman, A. (2023). The Philosophy of Probability Values Behaviour through Fractions and Composite Probability Function for Independent Events in the Discrete Case. European Scientific Journal, ESJ, 19(18), 1. https://doi.org/10.19044/esj.2023.v19n18p1
23. Justicz, J., Scheinerman, E. R., & Winkler, P. M. (1990). Random Intervals. The American Mathematical Monthly, 97(10), 881–889. https://doi.org/10.2307/2324324
24. Kane, Joseph W. & Sternheim, Morton M. (1988). Physics. Third Edition. Singapore: John Wiley & Sons,Inc.
25. Keller, J. B. (1986). The Probability of Heads. The American Mathematical Monthly, 93(3), 191–197. https://doi.org/10.2307/2323340
26. Klartag, B., Vershynin, R. Small ball probability and Dvoretzky’s Theorem. Isr. J. Math. 157, 193–207 (2007). Doi.org/10.1007/s11856-006-0007-1
27. Laning, Halcombe Jr., Battin, Richard H. (1956). Random Processes in Automatic Control. New York: McGraw-Hill Book Company, Inc.
28. Laplace, P. S. (1995). Théorie Analytique Des Probabilités. Quatrième & Troisième Édition. Paris: Éditions Jacques Gabay.
29. Latala, R., & Oleszkiewicz, K. (2005). Small ball probability estimates in terms of widths Studia Math.
30. Lebesgue, Henri. (1902). Intégrale, Longueur, Aire. Annali di Matematica, Serie III 7, 231-359. Doi.org/10.1007/BF02420592.
31. Lebesgue, Henri. (1989). Leçons sur L’intégration et La Recherche des Fonctions Primitives. Deuxième Édition. Paris: Éditions Jacques Gabay.
32. Lévy; Paul. (2004,2006). Calcul Des Probabilités. Paris: Éditions Jacques Gabay.
33. Loève, M. (1977). Probability theory, Volume I. 4th Edition. New York: Springer-Verlag.
34. Loève, M. (1978). Probability theory, Volume II. 4th Edition. New York: Springer-Verlag.
35. Mcginty, M. (2004). Geometric Probability for The Space-Time Plane. Pi Mu Epsilon Journal, 12(1), 25–35. http://www.jstor.org/stable/24340793
36. Meyer, P. L. (1970). Introductory Probability and StatisticalApplications. Second Edition. USA: Addison-Wesley Publishing Company, Inc.
37. Monticino, M. (2001). How to Construct a Random Probability Measure. International Statistical Review / Revue Internationale de Statistique, 69(1), 153–167. https://doi.org/10.2307/1403534
38. Parzen, E. (1960). Modern Probability Theory and Its Applications. New York, NY: John Wiley & Sons, Inc.
39. Parzen, E. (1962). Stochastic Processes. USA: Holden-Day, Inc.
40. Rényi, Alfréd. (1970). Probability Theory. Amsterdam: Holden-Day and Akadémiai Kiadó.
41. Ross, Sheldon M. (1989). A First Course in Probability. Third Edition. New York: Macmillan Publishing Company.
42. Shiryaev, Albert N. (2016). Probability-1. Third Edition. New York: Springer.
43. Stoll, Manfred. (1997). Introduction to Real Analysis. Boston: Addison Wesley Longman Inc.
44. Sveshnikov, A. A. (Ed.). (1968). Problems in Probability Theory, Mathematical Statistics and Theory of Random Functions. USA: W. B. Saunders Company.
45. Swokowski, Earl W. (1988). Calculus with Analytic Geometry. Second Alternate Edition. USA: PWS-KENT Publishing Company.
46. Szekely, Gabor. (2005). Half of a coin: Negative probabilities. Wilmott Magazine. 50. 66-68.
47. Tucker, Howard G. (1967). A Graduate Course in Probability. New York: Academic Press, Inc.
48. Uspensky, J. V. (1937). Introductions to Mathematical Probability. First Edition, Second Impression. New York: McGraw-Hill Book Company,Inc.
49. Velleman, D.J. (1997). Characterizing Continuity. Amer, Math, Monthly 104, 318-322.
Published
2024-01-23
How to Cite
Jughaiman, A. (2024). The Philosophy of Probability Values Behaviour, through Fractions and Composite Probability Function in the Continuous Case. European Scientific Journal, ESJ, 25, 572. Retrieved from https://eujournal.org/index.php/esj/article/view/17701
Section
ESI Preprints
Copyright (c) 2024 Abdulaziz Jughaiman
This work is licensed under a Creative Commons Attribution 4.0 International License.
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