Probability Theory II: Master the Concepts of Stochastic Processes and Statistical Inference

Question 1

Which of the following violates a property of a probability mass function (PMF)?

Accepted Answer

The PMF can take on values greater than 1.

Answer

The sum of probabilities over all possible outcomes is 1.

Answer

Probabilities are always non-negative.

Question 2

In a Markov chain with transition matrix P, what does the expression P^n represent?

Accepted Answer

The n-step transition matrix, giving the probabilities of transitioning from the current state to any other state in exactly n steps.

Answer

The probability of staying in the current state for n steps.

Answer

The probability of transitioning from the current state to the initial state in n steps.

Answer

The probability of transitioning from any state to the current state in n steps.

Question 3

Which of the following is not a type of stochastic process?

Accepted Answer

Linear regression

Answer

Markov process

Answer

Poisson process

Answer

Brownian motion

Question 4

In the study of Markov Chains, which concept is central?

Accepted Answer

Conditional probabilities

Answer

Independent probabilities

Answer

Random variables

Answer

Hypothesis verification

Question 5

Which definition precisely describes a Markov Chain?

Accepted Answer

A model where the probability of the next state depends only on the current state.

Answer

A model where the probability of the next state depends on all previous states.

Answer

A model where the probability of the next state is equal for all states.

Answer

A model where the probability of the next state is independent of the current state.

Question 6

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

Accepted Answer

Linear Regression

Answer

Markov Chain

Answer

Poisson Process

Answer

Brownian Motion

Question 7

Which of the following is not a stochastic process?

Accepted Answer

Deterministic process

Answer

Markov chain

Answer

Poisson process

Answer

Random walk

Question 8

Which stochastic process is used to model the number of events occurring within a fixed time interval?

Accepted Answer

Poisson Process

Answer

Markov Chain

Answer

Brownian Motion

Answer

Wiener Process

Question 9

Which of the following best defines the Markov property?

Accepted Answer

The future evolution of a process depends only on its present state.

Answer

The conditional probability of an event is dependent on all past events.

Answer

The distribution of a process remains constant over time.

Answer

The standard deviation of a process is time-invariant.

Question 10

What is the primary application of the Kolmogorov-Chapman equation?

Accepted Answer

Calculating transition probabilities in a Markov process

Answer

Determining the average waiting time for a specific event

Answer

Finding the distribution of the time until a certain event occurs

Answer

Estimating the likelihood of infrequent occurrences

Question 11

Which of the following scenarios best exemplifies a Bayesian inference problem?

Accepted Answer

Predicting the future price of a stock using historical data

Answer

Estimating the probability of obtaining heads when flipping a fair coin

Answer

Testing the hypothesis that two populations have different means

Answer

Fitting a regression line to a set of observations

Question 12

Under regularity conditions, the asymptotic distribution of the maximum likelihood estimator (MLE) is:

Accepted Answer

Normal distribution

Answer

Student's t-distribution

Answer

Chi-squared distribution

Answer

Uniform distribution

Question 13

Which of the following is a critical assumption of the Central Limit Theorem?

Accepted Answer

The sample size is sufficiently large

Answer

The underlying population is normally distributed

Answer

The sample mean is normally distributed

Answer

The sample variance is known

Question 14

What is the probability density function of a standard normal distribution?

Accepted Answer

$φ(x) = 1/(√2π)e^(-x²/2)$

Answer

$φ(x) = 1/(2π)e^(-x²/2)$

Answer

$φ(x) = 1/πe^(-x²/2)$

Answer

$φ(x) = 1/√(2π)e^(-x²)$

Question 15

Which statistical test is used to test the significance of a difference between means?

Accepted Answer

Student's t-test

Answer

Chi-square test

Answer

Kolmogorov-Smirnov test

Answer

Fisher's exact test

Question 16

What is the expected value of a Poisson random variable with parameter λ?

Accepted Answer

λ

Answer

0

Answer

λ^2

Answer

1/λ

Question 17

Which of the following is NOT an assumption of the central limit theorem?

Accepted Answer

The underlying distribution is normal

Answer

The sample size is sufficiently large

Answer

The observations are independent

Answer

The mean and variance of the population are finite

Question 18

What is the distribution of the sample mean of a sample of size n from a normal distribution with mean μ and variance σ^2?

Accepted Answer

Normal(μ, σ^2/n)

Answer

Normal(μ, σ^2)

Answer

Normal(μ/n, σ^2/n^2)

Answer

Normal(μ/n, σ^2)

Question 19

Which statistical method employs Bayes' theorem to update beliefs?

Accepted Answer

Bayesian updating

Answer

Maximum likelihood estimation

Answer

Hypothesis testing

Answer

Chi-square goodness-of-fit test

Question 20

Let X be a random variable with probability density function f(x). What is the total probability of X taking any possible value?

Accepted Answer

1

Answer

0

Answer

-1

Answer

2

Question 21

Which of the following is a valid probability mass function (PMF) of a discrete random variable X?

Accepted Answer

∑ P(X = x) = 1 for all x

Answer

P(X = x) < 0 for all x

Answer

P(X = x) > 1 for all x

Answer

∑ P(X = x) = 0 for all x

Question 22

Let X and Y be two independent random variables. Which of the following is true about the covariance of X and Y?

Accepted Answer

The covariance of X and Y is always zero

Answer

The covariance of X and Y is always one

Answer

The covariance of X and Y is equal to the product of their means

Answer

The covariance of X and Y is equal to the product of their variances

Question 23

Let X be a random variable with mean μ and variance σ^2. What is the standardized random variable?

Accepted Answer

(X - μ) / σ

Answer

X - μ

Answer

X + μ

Answer

(X + μ) / σ

Question 24

Which of the following is a key assumption of the central limit theorem?

Accepted Answer

The sample size is large

Answer

The sample size is small

Answer

The population distribution is normal

Answer

The population distribution is symmetric

Question 25

What is the mean (expected value) of a Poisson distribution with parameter λ?

Accepted Answer

λ

Answer

λ^2

Answer

e^(-λ)

Answer

1/λ

Question 26

What is the probability density function (PDF) for the standard normal distribution?

Accepted Answer

(2π)^(-1/2) * exp(-x^2/2)

Answer

λ * e^(-λx)

Answer

e^(-x)

Answer

x * e^(-x^2/2)

Question 27

Which of the following is a statistical inference method used to test hypotheses about population parameters?

Accepted Answer

Hypothesis testing

Answer

Confidence interval estimation

Answer

Regression analysis

Answer

Analysis of variance

Question 28

What is the commonly used significance level (α) in hypothesis testing?

Accepted Answer

0.05

Answer

0.01

Answer

0.10

Answer

0.20

Question 29

What is the primary purpose of hypothesis testing in statistical inference?

Accepted Answer

To confirm or reject a claim about a population parameter

Answer

To estimate the value of a population parameter

Answer

To predict the value of a dependent variable

Answer

To identify patterns in data

Question 30

What is the key assumption of a Markov chain?

Accepted Answer

The probability of the next state depends only on the current state

Answer

The probability of the next state is independent of all past states

Answer

The probability distribution of the states changes over time

Answer

The Markov chain is reversible

Question 31

What is the purpose of a statistical hypothesis test?

Accepted Answer

To make a decision about a population parameter based on sample data.

Answer

To estimate the mean of a random variable.

Answer

To compare two or more populations.

Answer

To model the distribution of a random variable.

Question 32

The p-value for a statistical hypothesis test is 0.03. Assuming a significance level of 0.05, what is the conclusion of the test?

Accepted Answer

Reject the null hypothesis.

Answer

Fail to reject the null hypothesis.

Answer

The test is inconclusive.

Answer

More data is required.

Question 33

Which of the following is NOT a measure of variability?

Accepted Answer

Mean

Answer

Range

Answer

Variance

Answer

Skewness

Question 34

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

Accepted Answer

Hypothesis testing

Answer

Markov chain

Answer

Poisson process

Answer

Brownian motion

Question 35

What is the probability of getting a 'head' in a single fair coin toss?

Accepted Answer

0.5

Answer

0.25

Answer

0.33

Answer

0.16

Question 36

What is the expected value of a Poisson random variable with parameter λ?

Accepted Answer

λ

Answer

λ^2

Answer

√λ

Answer

1/λ

Question 37

What is the concept of Brownian motion?

Accepted Answer

A random process that models the movement of particles in a fluid

Answer

A statistical model for the distribution of stock prices

Answer

A mathematical strategy for solving complex equations

Answer

A type of Markov chain with continuous states

Question 38

What is the difference between a hypothesis test and a confidence interval?

Accepted Answer

A hypothesis test determines whether a hypothesis is true or false, while a confidence interval provides an estimate of the true value with a certain level of confidence

Answer

A confidence interval is more precise than a hypothesis test

Answer

A hypothesis test is more robust than a confidence interval

Answer

There is no difference between a hypothesis test and a confidence interval

Question 39

What is the purpose of Bayesian inference?

Accepted Answer

To combine prior knowledge with data to make inferences about unknown parameters

Answer

To provide a frequentist interpretation of probability

Answer

To predict future events with certainty

Answer

To test statistical hypotheses

Question 40

What is the probability mass function of a Poisson distribution used in Probability Theory II?

Accepted Answer

$$ P(X = k) = rac{e^-λλ^k}{k!} $$

Answer

$$ P(X = k) = rac{1}{k!} $$

Answer

$$ P(X = k) = λ^k $$

Answer

$$ P(X = k) = λ^-k $$

Question 41

Which of the following is a characteristic of a Markov chain in Probability Theory II?

Accepted Answer

Memoryless property

Answer

Independent states

Answer

Normal distribution of transitions

Answer

Poisson distribution of jumps

Question 42

What is the Chapman-Kolmogorov equation used to calculate in Probability Theory II for a Markov chain?

Accepted Answer

$$ P_{ij}(n+m) = ∑_k P_{ik}(n) P_{kj}(m) $$

Answer

$$ P_{ij}(n+m) = P_{ij}(n) + P_{ij}(m) $$

Answer

$$ P_{ij}(n+m) = P_{ij}(n) * P_{ij}(m) $$

Answer

$$ P_{ij}(n+m) = P_{ji}(n) + P_{ij}(m) $$

Question 43

What is the role of the Fisher information matrix in maximum likelihood estimation in Probability Theory II?

Accepted Answer

It provides an estimate of the variance-covariance matrix of the parameter estimates

Answer

It determines the convergence rate of the estimation algorithm

Answer

It is used to calculate the likelihood-ratio test statistic

Answer

It is used to compute the confidence intervals for the parameter estimates

Question 44

Which of the following is not a type of stochastic process commonly used in probability modeling?

Accepted Answer

Logistic regression

Answer

Markov chain

Answer

Poisson process

Answer

Brownian motion

Question 45

Given a geometric random variable X with parameter p, what is its expected value denoted as E(X)?

Accepted Answer

p/(1-p)

Answer

1-p

Answer

p(1-p)

Answer

(1-p)/p

Question 46

In a Poisson process, which statistical distribution is typically used to model the time between consecutive events?

Accepted Answer

Exponential distribution

Answer

Normal distribution

Answer

Uniform distribution

Answer

Weibull distribution

Question 47

For an unbiased estimator, what is the relationship between its mean square error (MSE) and its variance?

Accepted Answer

MSE is equal to the variance.

Answer

MSE is twice the variance.

Answer

MSE is half the variance.

Answer

MSE is equal to the standard deviation.

Question 48

In a Poisson process with an arrival rate of 3 events per hour, what is the probability of observing exactly 5 events in a 2-hour interval?

Accepted Answer

0.279

Answer

0.140

Answer

0.374

Answer

0.474

Question 49

Which of the following is a key assumption of the central limit theorem?

Accepted Answer

The sample size is large enough.

Answer

The population distribution is normal.

Answer

The sample is taken with replacement.

Answer

The mean of the population is known.

Question 50

In a Markov chain, which of the following conditions implies that the chain is irreducible?

Accepted Answer

All states can be reached from any other state.

Answer

There exists a state that cannot be reached from any other state.

Answer

There exists a state that can be reached from all other states.

Answer

The chain has a finite number of states.

Question 51

Which of the following is a distinctive property of a Gaussian process?

Accepted Answer

It is fully specified by its mean and covariance functions.

Answer

It assumes a constant mean.

Answer

It is a non-stationary process.

Answer

It is a discrete-time process.

Question 52

Let X be a random variable following a binomial distribution with parameters n and p. What is the expected value of X?

Accepted Answer

np

Answer

n(1-p)

Answer

(1-p)n

Answer

0.5

Question 53

Which of the following is a continuous probability distribution?

Accepted Answer

Normal distribution

Answer

Binomial distribution

Answer

Poisson distribution

Answer

Negative binomial distribution

Question 54

What is the probability density function of a normal distribution with mean μ and variance σ^2?

Accepted Answer

1/(σ√(2π)) * exp(-(x-μ)^2/(2σ^2))

Answer

σ/(μ√(2π)) * exp(-(x-μ)^2/(2σ^2))

Answer

1/(μσ(2π)) * exp(-(x-μ)^2/(2σ^2))

Answer

σ/(μσ(2π)) * exp(-(x-μ)^2/(2σ^2))

Question 55

Which of the following is the method of moments estimator for the mean of a population?

Accepted Answer

Equating the sample mean to the population mean

Answer

Equating the sample variance to the population variance

Answer

Equating the sample median to the population median

Answer

Equating the sample mode to the population mode

Question 56

What is the purpose of a confidence interval?

Accepted Answer

To estimate the true population parameter with a specified level of confidence

Answer

To test the null hypothesis with a specified significance level

Answer

To determine the sample size needed for a given hypothesis test

Answer

To calculate the probability of observing a particular sample

Question 57

Which of the following is a measure of statistical dispersion?

Accepted Answer

Variance

Answer

Mean

Answer

Median

Answer

Mode

Question 58

What is the defining characteristic of a Markov chain?

Accepted Answer

Transition probabilities depend only on the current state

Answer

Guaranteed to reach a steady-state equilibrium

Answer

Unaffected by external random variables

Answer

Limited to a finite number of states

Question 59

What is a key assumption of the Central Limit Theorem?

Accepted Answer

The sample size is large enough.

Accepted Answer

The data comes from an independent random sample.

Answer

The population distribution is known and normal.

Answer

The sample mean is significantly different from the population mean.

Question 60

Which statistical test is non-parametric?

Accepted Answer

Wilcoxon rank-sum test

Answer

t-test

Answer

Pearson's correlation coefficient

Answer

Chi-squared test

Question 61

For a Bernoulli random variable X with parameter p = 0.6, what's its variance?

Accepted Answer

0.24

Answer

0.36

Answer

0.48

Answer

0.60

Question 62

In Bayesian inference with a prior distribution of p(θ) = Beta(α, β) and n data points from a Binomial distribution with parameter θ, what's the posterior distribution?

Accepted Answer

Beta(α + n, β + n)

Answer

Beta(α, β + n)

Answer

Beta(α + n, β)

Answer

Beta(α, β)

Question 63

What is the defining characteristic of a Markov Chain?

Accepted Answer

The future state depends only on the current state.

Answer

The future state is influenced by all past states.

Answer

The future state is independent of both the present and past states.

Question 64

In hypothesis testing, which symbol typically represents the null hypothesis?

Accepted Answer

H0

Answer

Hc

Answer

Ha

Answer

H1