MS-DS Master of Data science FREE MS-DS Master of Data science Statistical and Probabilistic Analysis Questions and Answers 2 — Questions and Answers
Question 1: In Bayesian inference, what does the posterior distribution represent?
- The likelihood of data given the parameter
- The updated belief about a parameter after observing data (Correct answer)
- The initial assumption before any data is collected
- The sampling distribution of a test statistic
Correct answer: The updated belief about a parameter after observing data
The posterior distribution combines the prior belief with the likelihood of observed data using Bayes' theorem to produce an updated belief about the parameter.
Question 2: Which property must a valid covariance matrix satisfy?
- All diagonal entries must be zero
- It must be positive semi-definite (Correct answer)
- It must have a determinant of exactly one
- All off-diagonal entries must be positive
Correct answer: It must be positive semi-definite
A valid covariance matrix must be symmetric and positive semi-definite, ensuring all variances and linear combinations of variables have non-negative variance.
Question 3: What does the Kolmogorov-Smirnov test primarily assess?
- Whether two population means are equal
- Whether a sample follows a specified distribution (Correct answer)
- Whether variances across groups are homogeneous
- Whether residuals are autocorrelated
Correct answer: Whether a sample follows a specified distribution
The Kolmogorov-Smirnov test compares the empirical distribution function of a sample against a reference cumulative distribution function to assess goodness of fit.
Question 4: If X and Y are independent random variables, what is Var(X - Y)?
- Var(X) - Var(Y)
- Var(X) + Var(Y) (Correct answer)
- Var(X) × Var(Y)
- Var(X) / Var(Y)
Correct answer: Var(X) + Var(Y)
For independent random variables, the variance of their difference equals the sum of their individual variances because independence means the covariance term is zero.
Question 5: In maximum likelihood estimation, what is the Fisher information used to approximate?
- The bias of the estimator
- The variance of the MLE asymptotically (Correct answer)
- The posterior probability of the parameter
- The p-value of a hypothesis test
Correct answer: The variance of the MLE asymptotically
The inverse of the Fisher information provides the asymptotic variance of the MLE, forming the basis of the Cramér-Rao lower bound.
Question 6: Which distribution arises as the sum of squared independent standard normal random variables?
- F-distribution
- Student's t-distribution
- Chi-squared distribution (Correct answer)
- Exponential distribution
Correct answer: Chi-squared distribution
The chi-squared distribution with k degrees of freedom is defined as the sum of k independent squared standard normal random variables.
