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