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