Stochastic Processes: Master the Fundamentals with Interactive Quizzes

Question 1

Which characteristic of a Markov chain specifies that future events are independent of the past, given the present state?

Accepted Answer

Memoryless future

Answer

Stationary distribution

Answer

Ergodicity

Answer

Irreducible states

Question 2

Which of the following time series models is characterized by a random walk with constant drift?

Accepted Answer

Wiener process

Answer

Autoregressive integrated moving average (ARIMA)

Answer

Generalised autoregressive conditional heteroskedasticity (GARCH)

Answer

Moving average (MA)

Question 3

If a Markov chain is irreducible and aperiodic, what is guaranteed about its long-run behavior?

Accepted Answer

Convergence to a unique stationary distribution

Answer

Periodic cycling between a fixed set of states

Answer

Transient behavior, never returning to the initial state

Answer

Convergence to a set of absorbing states

Question 4

Which theorem states that the sum of a large number of independent, identically distributed random variables approaches a normal distribution?

Accepted Answer

Central Limit Theorem

Answer

Markov Property

Answer

Poisson Distribution Theorem

Answer

Law of Averages

Question 5

What is the inter-arrival time distribution in a Poisson process?

Accepted Answer

Exponential

Answer

Normal

Answer

Gamma

Answer

Uniform

Question 6

What is the diffusion coefficient of a Brownian motion process?

Accepted Answer

σ^2

Answer

λ

Answer

μ

Answer

0

Question 7

Which fundamental property of Markov chains states that the probability of the next state depends solely on the current state?

Accepted Answer

Memorylessness

Answer

Stationarity

Answer

Ergodicity

Answer

Reversibility

Question 8

The Kolmogorov forward equation describes

Accepted Answer

Evolution of the probability distribution of a Markov process

Answer

Transition probabilities of a Markov chain

Answer

Expected value of a stochastic process

Answer

Variance of a stochastic process

Question 9

Which of the following is **not** an application of Brownian motion?

Accepted Answer

Predicting the weather

Answer

Modeling the movement of stock prices

Answer

Describing the diffusion of particles in a fluid

Answer

Simulating the motion of galaxies

Question 10

What is the mean of a stationary stochastic process?

Accepted Answer

The expected value over time

Answer

The expected value at any time t

Answer

The variance over time

Answer

The probability of the process being in a specific state

Question 11

What is the Chapman-Kolmogorov equation used for?

Accepted Answer

Calculating the probability of a transition between states in a Markov chain

Answer

Determining the mean of a Poisson process

Answer

Solving differential equations related to Brownian motion

Answer

Simulating a stationary stochastic process

Question 12

A stochastic process is defined as a set of random variables indexed by:

Accepted Answer

Time or space

Answer

Probability and distribution

Answer

Correlation and regression

Answer

Mean and variance

Question 13

In a Poisson process with rate λ, what is the probability of observing k events in a time interval of length t?

Accepted Answer

(λt)^k * e^(-λt) / k!

Answer

λt * e^(-λt)

Answer

(λt)^k * e^(-t)

Answer

e^(-λt) / k!

Question 14

What is the expected number of arrivals in a Poisson process with rate λ over a time interval of length t?

Accepted Answer

λt

Answer

λ/t

Answer

λt^2

Answer

t/λ

Question 15

The transition probability matrix of a first-order Markov chain is:

Accepted Answer

A square matrix where each element represents the probability of transitioning from one state to another

Answer

A diagonal matrix where each element represents the probability of staying in a state

Question 16

A key assumption of the Poisson process is that:

Accepted Answer

Events occur independently and at a constant average rate

Answer

The probability of an event occurring in a given time interval is proportional to the square of the length of the interval

Answer

Events occur in clusters

Question 17

Which of the following methods is used to simulate a Brownian motion process?

Accepted Answer

Euler-Maruyama method

Answer

Monte Carlo method

Answer

Rejection sampling

Answer

Importance sampling

Question 18

Which of the following is a defining characteristic of a Markov chain?

Accepted Answer

The probability of transitioning to the next state depends solely on the current state.

Answer

The process exhibits no memory.

Answer

The states are continuous.

Question 19

In a Poisson process, the probability of observing k events within a time interval of duration T is given by:

Accepted Answer

(e^-λT * λ^k) / k!

Answer

(e^-λT * λ^k) / (k-1)!

Answer

λ^k * e^-λT

Answer

(e^-λT * λ^k) / (k+1)!

Question 20

The transition matrix of a Markov chain with n states is a square matrix of order:

Accepted Answer

n x n

Answer

(n+1) x n

Answer

(n-1) x (n-1)

Answer

n x (n+1)

Question 21

For a Poisson random variable with parameter λ, the expected value is:

Accepted Answer

λ

Answer

1/λ

Answer

√λ

Answer

λ^2

Question 22

Which type of stochastic process models the diffusion of particles in a fluid?

Accepted Answer

Wiener process

Answer

Markov chain

Answer

Birth-death process

Question 23

The stationary distribution of a Markov chain is the distribution where:

Accepted Answer

The probabilities of states remain constant over time.

Answer

The initial distribution of the Markov chain.

Answer

The distribution with the highest probability.