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