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