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

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

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

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

A) Uniform distribution
B) Normal distribution
C) Bernoulli distribution
D) Exponential 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) Randomness decreases with more observations.
C) Expected values change with the number of observations.
D) Sample averages diverge from expected values.

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

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

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) No inference can be made about long-term behavior.
D) Behavior is completely random.

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

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

8. 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.

Created with That Quiz — where a math practice test is always one click away.