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

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

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

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

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

4. 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) Expected values change with the number of observations.
C) Randomness decreases with more observations.
D) Sample averages diverge from expected values.

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

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

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

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

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

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

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

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

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