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