Abstract
Probability is a concept familiar to the vast majority of readers on an intuitive level, however in a stricter sense it is generally poorly understood. A substantial fraction of this thesis draws ideas and tools from the rigorous theories of probability and thus it is apt to provide a fuller account of the rudimentary mathematical principles. This chapter does not aim to give an exhaustive exposition of probability theory (fuller discussion can be found in many existing texts, such as [2,5,10,11,16], but instead particular attention is given to the meaning of randomness, whereby a suitable parameterisation of the statistics involved can be formulated.
The scientist has a lot of experience with ignorance and doubt and uncertainty, and this experience is of very great importance, I think.Richard P. Feynman
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Although relative frequency, defined as \(f=\lim _{N\rightarrow \infty }n/N\), where \(n\) is the number of occurrences of a particular outcome in \(N\) repetitions of the random experiment, is perhaps the most intuitive manner in which to define probabilities, an axiomatic definition as first formulated by Kolmogorov [14, 15], is sometimes preferred since there is no guarantee that this limit exists.
- 2.
If this assumption of independence between disjoint time intervals is relaxed one enters the domain of quantum light sources, which possess different photon statistics. For example photon number state sources [6] emit a single photon at very precisely defined regular times.
- 3.
It would therefore be expected that a Poisson random variable with a large mean would be well approximated by a Gaussian random variable, a result that is born out in practise.
- 4.
It should be noted that the derivation given is not strictly rigorous since it only proves convergence of the partition function \(\exp [i \omega Z_n]\). The reader is directed to [17] for a fuller treatment.
References
N. Abramson, Information Theory and Coding (McGraw Hill, New York, 1963)
R.B. Ash, C.A. Doléans-Dade, Probability and Measure Theory (Elsevier Academic Press, New York, 1999)
M.L. Boas, Mathematical Methods in the Physical Sciences, 2nd edn. (Wiley, New York, 1983)
V. Delaubert, N. Treps, C. Fabre, A. Maître, H.A. Bachor, P. Réfrégier, Quantum limits in image processing. Europhys. Lett. 81, 44001 (2008)
W. Feller, Probability Theory and its Applications (Addison-Wesley, Reading, 1950)
M. Fox, Quantum Optics: An Introduction. Oxford Master Series in Physics (Oxford University Press, Oxford, 2006)
J.W. Goodman, Statistical Optics (Wiley, New York, 2004)
F. Hitoshi, A. Toshimitsu, N. Kunihiko, S. Yoshihisa, O. Takehiko, Blood flow observed by time-varying laser speckle. Opt. Lett. 10(3), 104–106 (1985)
N.E. Huang, Z. Shen, S.R. Long, M.C. Wu, H.H. Shih, Q. Zheng, N.-C. Yen, C.C. Tung, H.H. Liu, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. Math. Phys. Eng. Sci. 454, 903–995 (1998)
K. Itô, Introduction to Probability Theory (Cambridge University Press, Cambridge, 1984)
J. Jacod, P. Protter, Probability Essentials (Springer, New York, 2004)
S.M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, London, 1993)
A. Khintchine, Korrelationstheorie der stationären stochastischen Prozesse. Math. Ann. 109, 604–615 (1934)
A. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung (Springer, Heidelberg, 1933)
A. Kolmogorov, Foundations of the Theory of Probability, 2nd edn. (Chelsea Publishing, New York, 1956)
A. Leon-Garcia, Probability and Random Processes for Electrical Engineering (Addison-Wesley, Reading, 1994)
M. Loève, Probability Theory, 4th edn. (Springer, Heidelberg, 1978)
R. Loudon, The Quantum Theory of Light, 3rd edn. (Clarendon Press, Oxford, 2000)
L. Mandel, Fluctuations of Light Beams. Progress in Optics, vol. 2 (North-Holland Publishing Co., Amsterdam, 1963)
L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995)
W. Martin, P. Flandrin, Spectral analysis of nonstationary processes. IEEE Trans. Acoust. Speech. 33, 1461–1470 (1985)
G. Matz, F. Hlawatsch, W. Kozek, Generalized evolutionary spectral analysis and the Weyl spectrum of nonstationary random processes. IEEE Trans. Signal Process. 45, 1520–1534 (1997)
M.B. Priestley, Power spectral analysis of non-stationary random processes. J. Sound Vib. 6(1), 86–97 (1967)
L.L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, Reading, 1991)
C. E. Shannon, A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948)
C.E. Shannon, Communication in the presence of noise. Proc. IRE 37, 10–21 (1949)
N. Treps, V. Delaubert, A. Maître, J.M. Courty, C. Fabre, Quantum noise in multipixel image processing. Phys. Rev. A 71, 013820 (2005)
A. van den Bos, A Cramér-Rao lower bound for complex parameters. IEEE Trans. Signal Process. 42, 2859 (1994)
W. Weaver, Some Recent Contributions to the Mathematical Theory of Communication (University of Illinois Press, Urbana, 1949)
N. Wiener, Generalized harmonic analysis. Acta Math. 55, 117–258 (1930)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Foreman, M.R. (2012). Fundamentals of Probability Theory. In: Informational Limits in Optical Polarimetry and Vectorial Imaging. Springer Theses. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28528-8_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-28528-8_2
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-28527-1
Online ISBN: 978-3-642-28528-8
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)