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