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