Stochastic processes

1. Stochastic processes are mathematical objects that model random phenomena evolving over time or space. These processes are characterized by randomness and uncertainty in their behavior, making them essential tools in various fields such as statistics, finance, physics, and engineering. Unlike deterministic processes, stochastic processes involve probabilistic outcomes at each step, leading to a diverse range of possible outcomes. Key concepts in stochastic processes include random variables, probability distributions, Markov chains, and Brownian motion. Understanding and analyzing stochastic processes is crucial for making informed decisions in scenarios where randomness plays a significant role.

What is a stochastic process?

A) A constant value.
B) A linear equation.
C) A deterministic function.
D) A collection of random variables indexed by time or space.

2. What is the memoryless property of a stochastic process?

A) Future behavior does not depend on past history given the present.
B) The process always reverts back to its mean value.
C) Past behavior strongly influences future outcomes.
D) It exhibits periodic behavior.

3. Which distribution is commonly used to model arrival times in queuing systems?

A) Poisson distribution.
B) Normal distribution.
C) Weibull distribution.
D) Exponential distribution.

4. What is the Chapman-Kolmogorov equation in Markov chains?

A) An equation that calculates the stationary distribution directly.
B) An equation that describes the probability of transitioning between states in consecutive time steps.
C) An equation that predicts the long-term behavior of the chain.
D) An equation that models the uncertainty in transitions.

5. What is the state space of a stochastic process?

A) The historical record of past observations.
B) The set of all possible values that the process can take.
C) The fixed point of the process.
D) The set of future predictions.

6. What is the stationary distribution of a Markov chain?

A) A distribution with constantly changing parameters.
B) A distribution that depends on the initial state.
C) A probability distribution that remains unchanged over time.
D) A distribution that converges to zero over time.

7. What is the Wiener process also known as?

A) Poisson process.
B) Markov process.
C) Brownian motion.
D) Ornstein-Uhlenbeck process.

8. What is the autocovariance function of a stochastic process?

A) A measure of the periodicity of the process.
B) A measure of the dispersion of values around the mean.
C) A measure of the absolute difference between values.
D) A measure of the linear relationship between values at different time points.

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