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