- Which of the following is true about a stochastic process?
- (A) It is a deterministic process where future states are predictable.
- (B) It is a sequence of random variables indexed by time or space.
- (C) It always has a finite number of possible states.
- (D) It is a time-invariant process.
Answer: (B) It is a sequence of random variables indexed by time or space.
- Which of the following is an example of a Markov process?
- (A) A process where the future state depends only on the current state, not on the past states.
- (B) A process where the future state depends on all past states.
- (C) A process that is independent of the initial state.
- (D) A process that has no state transitions.
Answer: (A) A process where the future state depends only on the current state, not on the past states.
- What does the term “stationary process” refer to in the context of stochastic processes?
- (A) The process whose statistical properties do not change over time.
- (B) A process where the random variables are independent of each other.
- (C) A process whose outcomes are always the same.
- (D) A process whose distribution is uniform across all time intervals.
Answer: (A) The process whose statistical properties do not change over time.
- In a Poisson process, which of the following is true?
- (A) The events occur at a constant rate over time and are independent of each other.
- (B) The events are dependent on the previous event.
- (C) The event rate varies with time.
- (D) The process can only have one event at a time.
Answer: (A) The events occur at a constant rate over time and are independent of each other.
- Which of the following defines a Brownian motion process?
- (A) A process with continuous paths and independent, normally distributed increments.
- (B) A process where the next state is determined by a probability distribution.
- (C) A process that models the behavior of financial markets.
- (D) A process where the future state is deterministic.
Answer: (A) A process with continuous paths and independent, normally distributed increments.
- What is the key characteristic of a “Markov Chain”?
- (A) It has a finite number of states and a memoryless property.
- (B) The future state depends on both the current and all past states.
- (C) The states follow a deterministic sequence.
- (D) It is a continuous-time process.
Answer: (A) It has a finite number of states and a memoryless property.
- Which of the following is the primary feature of a “Gillespie algorithm”?
- (A) It is used to simulate discrete-event stochastic processes, especially in biochemical reactions.
- (B) It is a method to find the expected values of a stochastic process.
- (C) It is used to calculate the variance of a stochastic process.
- (D) It is a technique used to generate random walks.
Answer: (A) It is used to simulate discrete-event stochastic processes, especially in biochemical reactions.
- Which of the following is a characteristic of a Poisson process with rate λ?
- (A) The expected number of events in a fixed time interval is proportional to the length of the interval.
- (B) The probability of exactly two events occurring in a fixed time interval is constant.
- (C) The time between consecutive events follows a normal distribution.
- (D) The events are perfectly correlated.
Answer: (A) The expected number of events in a fixed time interval is proportional to the length of the interval.
- What is the difference between a “discrete-time” and a “continuous-time” stochastic process?
- (A) A discrete-time process has a continuous set of possible outcomes, while a continuous-time process has a finite set of outcomes.
- (B) A discrete-time process involves random variables indexed by continuous time, while a continuous-time process involves random variables indexed by discrete time.
- (C) In a discrete-time process, the state changes at specific time steps, while in a continuous-time process, the state can change at any time.
- (D) There is no difference between them.
Answer: (C) In a discrete-time process, the state changes at specific time steps, while in a continuous-time process, the state can change at any time.
- Which of the following describes a “stationary” and “ergodic” process?
- (A) The process’s statistical properties are time-varying but can be computed over time.
- (B) The process’s statistical properties do not change over time, and time averages equal ensemble averages.
- (C) The process has random fluctuations that follow a specific deterministic pattern.
- (D) The process’s mean is always zero.
Answer: (B) The process’s statistical properties do not change over time, and time averages equal ensemble averages.