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