Sample Questions and Answers
Which of the following best defines the concept of a measure in measure theory?
A) A rule that assigns values to sets
B) A rule for integrating functions
C) A concept for determining probabilities
D) A method to define probability distributions
Answer: A
In probability theory, if two events AAA and BBB are mutually exclusive, then:
A) P(A∩B)=0P(A \cap B) = 0P(A∩B)=0
B) P(A∪B)=1P(A \cup B) = 1P(A∪B)=1
C) P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B)P(A∪B)=P(A)+P(B)
D) P(A∩B)=P(A)+P(B)P(A \cap B) = P(A) + P(B)P(A∩B)=P(A)+P(B)
Answer: A
What does the CDF (Cumulative Distribution Function) of a random variable represent?
A) The probability that the random variable is greater than or equal to a specific value
B) The probability that the random variable is less than or equal to a specific value
C) The mean of the random variable
D) The variance of the random variable
Answer: B
Which of the following is true about the expectation E(X)E(X)E(X) of a random variable XXX?
A) It is always equal to zero
B) It represents the variance of the random variable
C) It is the weighted average of the possible values of the random variable
D) It represents the probability of occurrence of the random variable
Answer: C
In econometrics, which of the following is NOT a property of a well-behaved estimator?
A) Consistency
B) Efficiency
C) Bias
D) Unbiasedness
Answer: C
The law of large numbers (LLN) states that as the sample size nnn increases, the sample mean:
A) Converges in probability to the true population mean
B) Becomes more variable
C) Becomes less correlated with the true population mean
D) Converges in distribution to a normal distribution
Answer: A
The variance of a random variable XXX, denoted Var(X)\text{Var}(X)Var(X), is defined as:
A) E[X]E[X]E[X]
B) E[X2]−(E[X])2E[X^2] – (E[X])^2E[X2]−(E[X])2
C) E[X2]E[X^2]E[X2]
D) E[X]−XE[X] – XE[X]−X
Answer: B
In a continuous probability distribution, the probability that the random variable equals a specific value is:
A) 1
B) 0
C) Infinite
D) Dependent on the distribution
Answer: B
Which of the following is an assumption of the classical linear regression model?
A) Homoscedasticity
B) Non-zero mean error term
C) Non-independence of the error term
D) Linear relationship between the dependent and independent variables
Answer: A
Which of the following distributions is used in econometrics to describe errors in linear regression models when the errors are normally distributed?
A) Poisson distribution
B) Normal distribution
C) Exponential distribution
D) Binomial distribution
Answer: B
Which property of a random variable describes its central tendency?
A) Variance
B) Mean
C) Kurtosis
D) Standard deviation
Answer: B
In measure theory, the term sigma-algebra refers to:
A) A set of mutually exclusive events
B) A collection of sets closed under countable unions and intersections
C) A method to integrate probability functions
D) A concept used for computing conditional probabilities
Answer: B
In probability theory, Bayes’ theorem is used to:
A) Find the likelihood of independent events
B) Update the probability estimate for an event given new information
C) Calculate the cumulative distribution function
D) Determine the mean of a probability distribution
Answer: B
Which of the following is a valid property of a probability distribution?
A) The sum of probabilities of all possible outcomes equals 0
B) The sum of probabilities of all possible outcomes equals 1
C) The variance is always negative
D) The probability of a specific outcome is always greater than 1
Answer: B
What does the term “iid” (independent and identically distributed) refer to in the context of econometrics?
A) A set of variables that have the same distribution and are dependent on each other
B) A set of variables that are independent and come from the same probability distribution
C) A test to check the normality of the errors
D) A method to estimate the variance of the data
Answer: B
The “Central Limit Theorem” states that:
A) The sum of a large number of random variables will always be normally distributed
B) The sample mean approaches the population mean as the sample size increases
C) The distribution of sample means approaches a normal distribution as sample size increases
D) The variance of the sample means decreases with sample size
Answer: C
In statistical estimation, a confidence interval for a population parameter provides:
A) The exact value of the parameter
B) A range of values that are likely to contain the parameter with a given probability
C) The probability distribution of the parameter
D) The likelihood of observing a given sample mean
Answer: B
Which of the following best describes heteroscedasticity in regression models?
A) The variance of the error term is constant across observations
B) The error term has a non-zero mean
C) The variance of the error term is not constant across observations
D) The regression model does not include a constant term
Answer: C
The method of least squares in econometrics is used to:
A) Maximize the likelihood function
B) Minimize the sum of squared errors
C) Estimate the standard deviation of residuals
D) Calculate the probability of an event
Answer: B
Which of the following statements is true about the normal distribution?
A) It is always skewed to the right
B) The mean, median, and mode are all equal
C) The variance is always one
D) The distribution is bimodal
Answer: B
In measure theory, the Lebesgue integral is preferred over the Riemann integral because it:
A) Can handle discontinuous functions more effectively
B) Always converges faster
C) Requires fewer assumptions
D) Is easier to compute
Answer: A
A random variable XXX is said to follow a Poisson distribution if:
A) It models the number of events occurring in a fixed interval of time or space
B) Its mean and variance are equal
C) It describes the time between two events in a continuous process
D) It is used to model the spread of disease in a population
Answer: A
What is the primary goal of using econometric models?
A) To estimate the parameters of an economic theory
B) To describe the distribution of error terms
C) To test hypotheses about economic variables
D) To calculate confidence intervals for regression coefficients
Answer: A
The probability mass function (PMF) is used to describe:
A) Continuous random variables
B) Discrete random variables
C) The cumulative distribution function
D) The likelihood of a parameter
Answer: B
In a bivariate regression model, the coefficient of correlation rrr indicates:
A) The strength and direction of the linear relationship between two variables
B) The variance of the error terms
C) The mean of the dependent variable
D) The slope of the regression line
Answer: A
The term “stationarity” in time series analysis means:
A) The mean and variance of the time series are constant over time
B) The time series is always increasing
C) The time series has a constant skewness
D) The time series follows a random walk
Answer: A
In econometrics, endogeneity refers to:
A) The error term being correlated with an independent variable
B) A variable being exogenous to the model
C) The model having a non-linear relationship
D) The dependent variable being a function of time
Answer: A
The chi-squared test is used primarily to test:
A) The significance of regression coefficients
B) The goodness of fit of a model
C) The independence of two categorical variables
D) The normality of a distribution
Answer: C
What does the term “autocorrelation” refer to in econometrics?
A) The correlation between different variables in the model
B) The correlation of a variable with itself over time
C) The correlation of errors across different observations
D) The correlation of residuals with the fitted values
Answer: B
In measure theory, the concept of a measurable function means that:
A) The function is continuous and bounded
B) The function maps measurable sets to measurable sets
C) The function has a finite integral
D) The function is differentiable
Answer: B
Which of the following is an example of a random variable with a continuous probability distribution?
A) Number of heads in a coin toss
B) Time spent waiting for a bus
C) Number of customers arriving at a store
D) A roll of a fair die
Answer: B
In the context of econometrics, the Ordinary Least Squares (OLS) method assumes that the error terms have:
A) A mean of zero and constant variance
B) A non-zero mean
C) A normal distribution
D) A mean of one and a non-constant variance
Answer: A
In probability theory, the law of total probability states that:
A) The probability of an event is always 1
B) The total probability of an event can be broken down based on conditional events
C) The sum of the probabilities of all outcomes in an experiment is zero
D) Probabilities are independent of the events
Answer: B
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