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

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

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

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

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

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

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

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

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

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

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

7. What is the Wiener process also known as?

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

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

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

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