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