A) A process that remains constant over time. B) A random process evolving over time. C) A deterministic process with fixed outcomes. D) A process that only occurs in discrete steps.
A) Average value of the process over time. B) Maximum value the process can attain. C) Exact value of the process at a given time. D) Set of all possible values that the process can take.
A) Uniform distribution B) Normal distribution C) Bernoulli distribution D) Exponential distribution
A) Long-term average behavior can be inferred from a single realization. B) Short-term analysis is sufficient for understanding long-term behavior. C) No inference can be made about long-term behavior. D) Behavior is completely random.
A) Maximum correlation possible for the process. B) Average of the process over time. C) Exact form of the process at a given time. D) Measure of correlation between values at different time points.
A) Deterministic process B) Brownian motion C) Geometric process D) Markov process
A) Specifies the final state of the process. B) Calculates the average time spent in each state. C) Describes probabilities of moving to different states. D) Determines the initial state of the process.
A) Sample averages diverge from expected values. B) As the number of observations increases, sample averages converge to expected values. C) Expected values change with the number of observations. D) Randomness decreases with more observations.
A) Exclusively in mathematics and statistics. B) Biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, and telecommunications. C) Primarily in linguistics and anthropology. D) Only in finance and economics.
A) A. K. Erlang. B) Andrey Kolmogorov. C) Louis Bachelier. D) Albert Einstein.
A) It can only take integer values. B) The state space is the real line. C) The state space is finite. D) The index set consists of integers.
A) 1662 B) 1934 C) 1888 D) 1713
A) Joseph Doob B) Ladislaus Bortkiewicz C) Aleksandr Khinchin D) Jakob Bernoulli
A) Aleksandr Khinchin B) Andrei Kolmogorov C) Francis Edgeworth D) Joseph Doob
A) 17th century B) 18th century C) 16th century D) 14th century
A) Jakob Bernoulli B) Aleksandr Khinchin C) Ladislaus Bortkiewicz D) Andrei Kolmogorov
A) Ars Conjectandi B) De Motu Corporum C) Philosophiæ Naturalis Principia Mathematica D) Principia Mathematica
A) Latin word meaning 'chance' B) Old English word meaning 'luck' C) Middle French word meaning 'speed, haste' D) Greek word meaning 'to aim at a mark'
A) 1713 B) 1888 C) 1934 D) 1662
A) Andrei Kolmogorov B) Jakob Bernoulli C) Ladislaus Bortkiewicz D) Joseph Doob
A) {X_t}_{t∉T} B) X(t) C) {X(t)}_{t∈T} D) {X_t}
A) X(t) B) {X_t}_{t∈T} C) {X(t)}_{t∈T} D) {X_t}
A) p B) t C) 0.5 D) 1-p
A) An idealized coin flip B) A deterministic outcome C) A Poisson event D) A continuous distribution
A) 0.5 B) t C) 1-p D) p
A) {0, 1, 2, ...} B) (−∞, ∞) C) [0, ∞) D) [1, ∞)
A) Drawing cards from a deck B) Rolling a die C) Repeatedly flipping a coin D) Measuring time intervals
A) Zero B) t C) One D) p
A) 0 or 1 B) -1 or 0 C) Any real number D) +1 or -1
A) The integers B) Real numbers C) Rational numbers D) Natural numbers
A) Integers B) Complex numbers C) Real numbers D) The natural numbers
A) Albert Einstein B) Kiyoshi Itô C) Norbert Wiener D) Andrey Kolmogorov
A) Brownian motion B) Markov chain C) Poisson process D) Lévy flight
A) n-dimensional B) 3-dimensional C) 2-dimensional D) 1-dimensional
A) Thermodynamics B) Classical mechanics C) Quantitative finance D) Electromagnetism
A) Black–Scholes–Merton model B) CAPM model C) Modern portfolio theory D) Efficient market hypothesis
A) t1 > t2. B) t1 and t2 are independent. C) t1 ≤ t2. D) t1 = t2.
A) Probability measure. B) Set intersection. C) Union of sets. D) Function composition.
A) Finite-dimensional distributions. B) The second moment. C) The mean and variance. D) The index set.
A) An unordered set. B) No specific order. C) A partial order relation. D) A total order relation.
A) Markov property B) Stationarity C) Independence D) Continuity
A) 1931 B) 1907 C) 1928 D) 1912
A) Thorvald Thiele B) Albert Einstein C) Norbert Wiener D) Louis Bachelier
A) Continuous and differentiable at all points. B) Cumulative distribution function. C) Continue à droite, limite à gauche (right-continuous with left limits). D) Constant amplitude discrete linear graph.
A) 1920s B) 1950s C) 1960s D) 1900s
A) Differential equation B) Diffusion equation C) Fourier equation D) Least squares equation
A) R B) V C) C D) E
A) Paul Lévy B) Harald Cramér C) Andrei Kolmogorov D) Wolfgang Doeblin
A) Marian Smoluchowski B) Percy Daniell C) Albert Einstein D) Louis Bachelier
A) 1903 B) 1910 C) 1920 D) 1909
A) Émile Borel B) Sergei Bernstein C) Paul Lévy D) Andrei Kolmogorov
A) Equivalent B) Version C) Modification D) Stochastic equivalence
A) Poincaré B) Sydney Chapman C) Andrey Kolmogorov D) Maurice Fréchet
A) Paul Lévy B) Henri Lebesgue C) David Hilbert D) Andrei Kolmogorov
A) Sydney Chapman B) Louis Bachelier C) Andrey Kolmogorov D) Norbert Wiener
A) Thorvald Thiele B) Albert Einstein C) Norbert Wiener D) Louis Bachelier
A) Martin Hairer B) Gilbert Hunt C) Srinivasa Varadhan D) Wendelin Werner
A) A random variable. B) A sigma-algebra on Ω. C) An index set for time. D) A probability measure.
A) Paul Lévy B) Henri Lebesgue C) Sergei Bernstein D) Émile Borel
A) A dense countable subset. B) No specific properties. C) A finite number of elements. D) An uncountable number of elements.
A) 1960 B) 1970 C) 1953 D) 1945
A) Eugene Dynkin B) Andrey Kolmogorov C) Maurice Fréchet D) Poincaré
A) Gilbert Hunt B) Shizuo Kakutani C) Paul-André Meyer D) Kiyosi Itô
A) Paul Lévy B) Norbert Wiener C) Anatoliy Skorokhod D) Andrey Kolmogorov
A) S B) D C) C D) F
A) Josiah Gibbs B) Ludwig Boltzmann C) James Clerk Maxwell D) Rudolf Clausius
A) Sydney Chapman B) Paul Ehrenfest C) William Feller D) Louis Bachelier
A) Filip Lundberg B) Siméon Poisson C) A.K. Erlang D) Harry Bateman
A) Marian Smoluchowski B) Albert Einstein C) Percy Daniell D) Jean Perrin
A) 1950s B) 1900 C) 1880 D) 1912
A) Joseph Doob B) Kiyosi Itô C) Alexander Wentzell D) Gilbert Hunt
A) Grundbegriffe der Wahrscheinlichkeitsrechnung B) Introduction to Measure Theory C) Foundations of Probability Theory D) The Theory of Stochastic Processes
A) Deterministic models B) Linear models C) Non-linear models D) Stochastic models
A) Josiah Gibbs B) Rudolf Clausius C) James Clerk Maxwell D) Ludwig Boltzmann
A) Paul-André Meyer B) Gilbert Hunt C) Alexander Wentzell D) Srinivasa Varadhan
A) Phone calls B) Alpha particles C) Differential equations D) Insurance claims
A) Andrey Markov B) Maurice Fréchet C) Sydney Chapman D) Irénée-Jules Bienaymé
A) Jean Ville B) Kiyosi Itô C) Gilbert Hunt D) Joseph Doob
A) Independence and identical distribution B) Consistency conditions C) Linearity and continuity D) Normality and stationarity
A) Joseph Doob B) Shizuo Kakutani C) Sergei Bernstein D) Gilbert Hunt
A) Schramm–Loewner evolution B) Potential theory C) Stochastic calculus D) Theory of large deviations
A) 1928 B) 1932 C) 1934 D) 1937
A) Gambler's ruin B) Brownian motion C) Point process D) Renewal process
A) The development of calculus. B) The invention of algebra. C) The study of geometry. D) A gambling problem.
A) They require a dense countable subset of their index set to be separable. B) Their separability depends on the state space S. C) They are always separable. D) They cannot be separable.
A) 1945 B) 1925 C) 1929 D) 1933
A) Jacob Bernoulli B) Karl Pearson C) Christiaan Huygens D) George Pólya
A) Andrei Kolmogorov B) Harald Cramér C) Joseph Doob D) William Feller
A) Jean Perrin B) Leonard Savage C) Thorvald Thiele D) Percy Daniell
A) Physics B) Measure theory C) Financial mathematics D) Time-series analysis
A) Orthogonality implies independence. B) They are unrelated concepts. C) Independence implies uncorrelatedness. D) Uncorrelatedness implies independence.
A) Leonard Savage B) Louis Bachelier C) Marian Smoluchowski D) Norbert Wiener
A) 1905 B) 1919 C) 1930s D) 1713
A) Sergei Bernstein B) Gilbert Hunt C) Joseph Doob D) Kiyosi Itô
A) Paul Lévy B) Joseph Doob C) Andrey Kolmogorov D) Norbert Wiener
A) Solving deterministic differential equations B) Markov chain Monte Carlo methods in Bayesian statistics C) Simulating non-random objects D) Analyzing linear regression models
A) The Great Depression B) The Cold War C) The Russian Revolution D) World War II
A) Statistical mechanics B) Thermodynamics C) Classical mechanics D) Quantum mechanics
A) Itô's lemma B) Lévy's continuity theorem C) Central Limit Theorem D) Kolmogorov's existence theorem
A) 1912 B) 1931 C) 1906 D) 1928 |