Stochastic Processes: Master Randomness in Mathematics

Question 1

In Markov chain theory, the probability of transitioning to a future state depends solely on:

Accepted Answer

The current state

Answer

The previous state

Answer

The initial state

Answer

All previous states

Question 2

Which of the following accurately describes Brownian motion?

Accepted Answer

Continuous trajectories, independent increments, and normally distributed increments

Answer

Continuous trajectories, independent increments, and Poisson distributed increments

Answer

Discontinuous trajectories, independent increments, and normally distributed increments

Answer

Discontinuous trajectories, independent increments, and Poisson distributed increments

Question 3

The infinitesimal mean of a Poisson process with intensity λ is:

Accepted Answer

λ

Answer

λ * t

Answer

t

Answer

λ / t

Question 4

A regular Markov chain is characterized by:

Accepted Answer

All states communicating with each other

Answer

All states being transient

Answer

All states being recurrent

Answer

The absence of absorbing states

Question 5

In a Markov chain with transition matrix P, what represents the probability of a transition from state i to state j?

Accepted Answer

P(i, j)

Answer

P(j, i)

Answer

P(i, i) * P(j, j)

Answer

P(i, j) * P(j, i)

Question 6

Customers arrive at a bank according to a Poisson process with parameter λ. What is the probability that at least two customers arrive within the next 2 units of time?

Accepted Answer

1 - 2λ * e^(-2λ)

Answer

1 - e^(-2λ)

Answer

e^(-2λ)

Answer

2λ * e^(-2λ)

Question 7

Doubling the intensity of a Poisson process affects the expected inter-arrival time by:

Accepted Answer

Halving it

Answer

Doubling it

Answer

Leaving it unchanged

Answer

Quadrupling it

Question 8

In a Poisson process with parameter λ over a time interval of length t, what is the variance of the number of arrivals?

Accepted Answer

λ * t

Answer

t / λ

Answer

√(λ * t)

Answer

λ / t

Question 9

The transition probability of a Markov chain is defined by which equation?

Accepted Answer

P(X_n = y | X_0 = x) = P(X_n = y | X_{n-1} = x)

Answer

P(X_n = y | X_0 = x) = P(X_n = y)

Answer

P(X_n = y | X_0 = x) = P(X_{n-1} = x | X_n = y)

Answer

P(X_n = y | X_0 = x) = P(X_0 = x | X_n = y)

Question 10

The mean of a Poisson process with rate λ is:

Accepted Answer

λ

Answer

1/λ

Answer

√λ

Answer

e^λ

Question 11

Which real-world scenario exemplifies an application of Markov chains?

Accepted Answer

Modeling customer behavior within a retail store.

Answer

Predicting stock market trends.

Answer

Analyzing genetic sequences.

Answer

Simulating disease outbreaks.

Question 12

A distinguishing characteristic of a martingale is that:

Accepted Answer

Its expected future value equals its current value.

Answer

Its increments are mutually independent.

Answer

It follows a normal distribution.

Answer

It is a submartingale.

Question 13

The variance of a Brownian motion process with drift μ and diffusion coefficient σ is:

Accepted Answer

σ^2 * t

Answer

μ^2 * t

Answer

σ * t

Answer

μ * t

Question 14

The Chapman-Kolmogorov equations primarily describe:

Accepted Answer

The transition probabilities of a Markov chain.

Answer

The conditional probabilities of a Poisson process.

Answer

The moments of a Brownian motion process.

Answer

The stationary distribution of a birth-death process.

Question 15

For a renewal process with inter-renewal times X, what is the mean time until the first renewal?

Accepted Answer

EX

Answer

E(X^2)/2

Answer

Var(X)/EX

Answer

EX/(EX)^2

Question 16

Which of the following is NOT a characteristic of a discrete-time Markov chain?

Accepted Answer

The chain has independent increments.

Answer

The state of the chain at time t+1 depends only on the state at time t.

Answer

The transition probabilities are constant over time.

Answer

The number of states is finite.

Question 17

What is the expected number of transitions a discrete-time Markov chain will make from state i to state j in n steps?

Accepted Answer

n * p_ij

Answer

n^2 * p_ij

Answer

sqrt(n) * p_ij

Answer

1 / (n * p_ij)

Question 18

What is the variance of the number of events that occur in a Poisson process with rate λ in a time interval of length t?

Accepted Answer

λ * t^2

Answer

λ * t

Answer

sqrt(λ * t)

Answer

1 / (λ * t)

Question 19

What is the probability density function of a Brownian motion process with drift μ and diffusion coefficient σ?

Accepted Answer

(1 / sqrt(2πt)) * exp(-(x - μt)^2 / (2σ^2t))

Answer

(1 / sqrt(2πt^2)) * exp(-(x - μt)^2 / (2σ^2t^2))

Answer

(1 / sqrt(2πt)) * exp(-(x - σt)^2 / (2μ^2t))

Answer

(1 / sqrt(2πt^2)) * exp(-(x - σt)^2 / (2μ^2t^2))

Question 20

What is the expected time until the first event occurs in a Poisson process with rate λ?

Accepted Answer

1 / λ

Answer

λ

Answer

1 / sqrt(λ)

Answer

sqrt(λ)

Question 21

In a Poisson process with rate λ, what is the expected number of arrivals within a time interval of length t?

Accepted Answer

λt

Answer

λ

Answer

λ^2t

Answer

e^(-λt)

Question 22

In a Markov chain, what is the probability of transitioning from state A to state B given the transition matrix P?

Accepted Answer

P(A, B)

Answer

P(B, A)

Answer

P(A)*P(B)

Answer

P(A)+P(B)

Question 23

What does the stationary distribution of a Markov chain represent?

Accepted Answer

The distribution of the chain in the long run

Answer

The distribution of the chain at any given time

Answer

The distribution of transition probabilities

Answer

The distribution of the state space

Question 24

Which of the following is a core concept in stochastic processes?

Accepted Answer

Markov chains

Answer

Differential equations

Answer

Linear algebra

Answer

Newtonian physics

Question 25

The expected value of a Brownian motion process is:

Accepted Answer

0

Answer

Time (t)

Answer

Variance (σ^2)

Answer

Variance multiplied by time (σ^2t)

Question 26

What is the mathematical representation of a Markov chain with n states?

Accepted Answer

Transition matrix: (a_ij)

Answer

Identity matrix: (a_ii)

Answer

Adjacency matrix: (a_ji)

Answer

Difference matrix: (a_ij) - (a_ji)

Question 27

For a Poisson process, calculate the probability of observing exactly k events within a time interval of length t:

Accepted Answer

e^(-λt) * (λt)^k / k!

Answer

1 - e^(-λt) * (λt)^k / k!

Answer

λt * e^(-λt) * (λt)^k / k!

Answer

Insufficient information provided

Question 28

Determine the anticipated number of crossings through a specified level by a Brownian motion process over a time interval of length T:

Accepted Answer

T/(2πσ^2)

Answer

2πσ^2/T

Answer

σ^2/T

Answer

T/σ^2

Question 29

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

Accepted Answer

The future state depends only on the present state

Answer

The state space is continuous

Answer

The transition probabilities change over time

Answer

The process is deterministic

Question 30

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

t/λ

Answer

1/λt

Question 31

What is the probability density function of a Brownian motion process?

Accepted Answer

Normal distribution

Answer

Exponential distribution

Answer

Poisson distribution

Answer

Beta distribution

Question 32

What is the expected time it takes for a Poisson process with rate λ to have its first arrival?

Accepted Answer

1/λ

Answer

λ

Answer

1/λ^2

Answer

λ^2

Question 33

Identify the process that does NOT fall under the category of stochastic processes:

Accepted Answer

Deterministic process

Answer

Poisson process

Answer

Markov chain

Question 34

A Markov chain, a particular type of stochastic process, is defined by which defining property?

Accepted Answer

Memorylessness

Answer

Ergodicity

Answer

Independence

Answer

Stationarity

Question 35

The Poisson process, another fundamental stochastic process, is characterized by its ability to model which aspect of random events?

Accepted Answer

The number of occurrences within a fixed time interval

Answer

The distribution of occurrences over time

Answer

The time span between occurrences

Question 36

Brownian motion, a well-known stochastic process, finds applications in modeling which phenomenon?

Accepted Answer

Random movement of particles

Answer

Fluid dynamics

Answer

Population growth

Answer

Spread of infectious diseases

Question 37

Stochastic processes have gained prominence due to their extensive applications in various fields. Which of the following is an example of a real-world application of stochastic processes?

Accepted Answer

Weather forecasting

Answer

Integral calculations

Answer

Solving complex equations

Answer

Polynomial root finding

Question 38

Stochastic processes, despite their wide-ranging applications, also come with certain limitations. Which of the following is a recognized limitation of stochastic processes?

Accepted Answer

Inherent uncertainty in predictions

Answer

Computational simplicity

Answer

Highly accurate results

Answer

Extensive range of applications

Question 39

In a Poisson process with rate lambda, what is the expected waiting time until the occurrence of the first event?

Accepted Answer

1/lambda

Answer

lambda

Answer

2lambda

Answer

1/2lambda

Question 40

Which of the following is NOT a characteristic of a Markov chain?

Accepted Answer

The probability of transitioning from one state to another depends on all previous states.

Answer

The process has no memory.

Answer

It is a discrete-time stochastic process with a finite state space.

Question 41

What is the probability of getting exactly 5 heads in 10 flips of a fair coin, according to the binomial distribution?

Accepted Answer

0.246

Answer

0.01

Answer

0.5

Answer

0.1

Question 42

Which of the following is a property of Brownian motion?

Accepted Answer

It is a continuous-time stochastic process with continuous sample paths.

Answer

It has a constant mean.

Answer

It is a discrete-space stochastic process.

Answer

It is a stationary process.

Question 43

Which of the following is an application of Markov chains?

Accepted Answer

Modeling queuing systems in real-world scenarios.

Answer

Solving differential equations.

Answer

Finding the roots of polynomials.

Answer

Calculating the area under a curve.

Question 44

What is the variance of Brownian motion with a drift of 2?

Accepted Answer

4t

Answer

2t

Answer

t

Question 45

Which of the following is a limit theorem for stochastic processes?

Accepted Answer

Central Limit Theorem.

Answer

Pythagorean Theorem.

Answer

Fourier Theorem.

Answer

Binomial Theorem.

Question 46

What is the expected time until the first arrival in a Poisson process with a rate of 3 per minute?

Accepted Answer

0.33 minutes

Answer

3 minutes

Answer

10 minutes

Answer

1 minute

Question 47

Which of the following is NOT a continuous-time stochastic process?

Accepted Answer

Markov chain

Answer

Poisson process

Answer

Brownian motion

Answer

Geometric process

Question 48

The mean of a Poisson process with rate λ is:

Accepted Answer

λt

Answer

λ

Answer

λt^2

Question 49

The variance of a Poisson process with rate λ is:

Accepted Answer

λt

Answer

1/λt

Answer

λ

Question 50

The infinitesimal generator of a Markov chain is a matrix that:

Accepted Answer

Describes the transition rates between states.

Answer

Describes the probabilities of remaining in each state.

Answer

Describes the expected time spent in each state.

Question 51

Brownian motion is a stochastic process that:

Accepted Answer

Describes the random movement of particles in a fluid.

Answer

Describes the waiting time between arrivals in a queue.

Answer

Describes the growth of a population over time.

Question 52

The probability density function of a normal random variable has which of the following shapes?

Accepted Answer

A bell-shaped curve

Answer

A straight line

Answer

A parabolic curve

Answer

An exponential curve