Abstract
This Chapter builds on Kolmogorov’s axioms and reviews basic concepts of probability theory. Mathematical description and analysis of stochastic processes with emphasis on several relevant special classes like stationary and ergodic processes are explained. Some special processes such as the normal process, the Wiener process, the Markov process and white noise are then discussed.
Probability is expectation founded upon partial knowledge. A perfect
acquaintance with all the circumstances affecting the occurrence of
an event would change expectation into certainty, and leave neither
room nor demand for a theory of probabilities.
— George Boole
If we have an atom that is in an excited state and so is going to emit
a photon, we cannot say when it will emit the photon. It has a certain
amplitude to emit the photon at any time, and we can predict only a
probability for emission; we cannot predict the future exactly.
— Richard Feynman
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Notes
- 1.
Girolamo Cardano, Italian Renaissance mathematician (1501–1576); famous for his solution of the cubic equation and as the inventor of the universal joint.
- 2.
Galileo Galilei, Italian astronomer, physicist, engineer, mathematician and philosopher (1564–1642); universally considered to be the father of observational astronomy and more significantly of the scientific method.
- 3.
Pierre de Fermat, French lawyer and mathematician (1607–1665); probably best known for his “last theorem” which is very easy to state but which resisted proof until the end of the 20th century.
- 4.
Blaise Pascal, French mathematician, physicist and philosopher (1623–1662).
- 5.
Jakob Bernoulli, Swiss mathematician, (1654–1705).
- 6.
Andrey Nikolaevich Kolmogorov, Russian mathematician, (1903–1987).
- 7.
See Subsection 3.3.8 on Normal Distribution.
- 8.
Pafnuty Lvovich Chebyshev, Russian mathematician (1821–1894); famous for his work on probability and number theory; also as the doctoral advisor of Alexandr Mikhailovich Lyapunov and Andrei Andreyevich Markov.
- 9.
Alexandr Mikhailovich Lyapunov, Russian mathematician (1857–1918).
- 10.
Robert Brown, Scottish botanist (1773 – 1858).
- 11.
Norbert Wiener, American mathematician, control engineer (1894–1964); famous as the “father of cybernetics”.
- 12.
Symmetry conditions: if \(\{i_1, i_2, ..., i_j\}\) is a permutation of \(\{1, 2, ..., j\}\), then \(F(t_{i_1}, t_{i_2}, ..., t_{i_j}, x_{i_1}, x_{i_2}, ..., x_{i_j}) = F(t_1, t_2, ..., t_j, x_1, x_2, ..., x_j)\). Compatibility conditions are \(F(t_1, ..., t_i, t_{i+1}, ..., t_j, x_1, ..., x_i, ...) = F(t_1, ..., t_i, x_1, ..., x_i)\).
- 13.
This is because the correlation function \(R(t,s)= R(t+(-s), s+(-s))=R(t-s, 0)\) and \(C(t,s)= C(t+(-s), s+(-s))=C(t-s, 0)\).
- 14.
Salomon Bochner, American mathematician (1899–1982).
- 15.
Andrey Andreyevich Markov, Russian mathematician (1856–1922); famous for his work on stochastic processes.
- 16.
Thomas Bayes, English mathematician and statistician, (1701–1761); famous for formulating the conditional probability theorem named after him.
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Hacιsalihzade, S.S. (2018). Probability and Stochastic Processes. In: Control Engineering and Finance. Lecture Notes in Control and Information Sciences, vol 467. Springer, Cham. https://doi.org/10.1007/978-3-319-64492-9_3
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DOI: https://doi.org/10.1007/978-3-319-64492-9_3
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