Stochastic process

1. A stochastic process is a mathematical object consisting of a collection of random variables, typically indexed by time. It represents the evolution of some system over time where uncertainty or randomness is involved in the system's behavior. Stochastic processes are used in various fields such as finance, physics, biology, and engineering to model random phenomena and analyze their properties. These processes can be classified into different types based on their properties, such as discrete-time or continuous-time, stationary or non-stationary, and Markovian or non-Markovian, providing a powerful framework for studying and understanding complex systems influenced by randomness.

What is a stochastic process?

A) A process that only occurs in discrete steps.
B) A deterministic process with fixed outcomes.
C) A process that remains constant over time.
D) A random process evolving over time.

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

A) Average value of the process over time.
B) Maximum value the process can attain.
C) Exact value of the process at a given time.
D) Set of all possible values that the process can take.

3. In a Poisson process, what is the inter-arrival time distribution?

A) Exponential distribution
B) Normal distribution
C) Uniform distribution
D) Bernoulli distribution

4. What is the autocorrelation function of a stochastic process?

A) Measure of correlation between values at different time points.
B) Exact form of the process at a given time.
C) Average of the process over time.
D) Maximum correlation possible for the process.

5. Which of the following is NOT a type of stochastic process?

A) Deterministic process
B) Markov process
C) Geometric process
D) Brownian motion

6. What does ergodicity imply in the context of stochastic processes?

A) Long-term average behavior can be inferred from a single realization.
B) Behavior is completely random.
C) Short-term analysis is sufficient for understanding long-term behavior.
D) No inference can be made about long-term behavior.

7. What is the Law of Large Numbers in the context of stochastic processes?

A) As the number of observations increases, sample averages converge to expected values.
B) Randomness decreases with more observations.
C) Sample averages diverge from expected values.
D) Expected values change with the number of observations.

8. What is the role of a transition matrix in a Markov chain?

A) Describes probabilities of moving to different states.
B) Calculates the average time spent in each state.
C) Specifies the final state of the process.
D) Determines the initial state of the process.

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