MS-DS Master of Data science Statistical and Probabilistic Analysis 3 — Questions and Answers
Question 1: In Bayesian inference, what does the posterior distribution represent?
- Updated belief about a parameter after observing data (Correct answer)
- The probability distribution of the data before any observation
- The likelihood function normalized over parameters
- Prior belief multiplied by sample size
Correct answer: Updated belief about a parameter after observing data
The posterior is proportional to the likelihood times the prior, representing updated belief after incorporating observed data.
Question 2: Which distribution arises as the ratio of a standard normal to the square root of a chi-squared variable divided by its degrees of freedom?
- Student's t-distribution (Correct answer)
- F-distribution
- Beta distribution
- Gamma distribution
Correct answer: Student's t-distribution
The t-distribution is defined as Z / √(χ²/ν), where Z is standard normal and χ² has ν degrees of freedom.
Question 3: A data scientist applies a log transformation to a right-skewed variable before regression. What is the primary statistical reason?
- To reduce skewness and stabilize variance (Correct answer)
- To increase the range of the variable
- To remove outliers automatically
- To convert discrete data to continuous
Correct answer: To reduce skewness and stabilize variance
Log transformation compresses large values, reducing right skew and often stabilizing variance (homoscedasticity).
Question 4: The moment generating function (MGF) of a random variable X is M(t) = eˢᵗ⁺⁽ˢ²ᵗ²/²⁾. Which distribution does X follow?
- Normal distribution (Correct answer)
- Exponential distribution
- Poisson distribution
- Uniform distribution
Correct answer: Normal distribution
The MGF of a Normal(μ, σ²) distribution is e^(μt + σ²t²/2), matching the given form with μ=s and σ²=s².
Question 5: In a one-way ANOVA with 4 groups, what are the degrees of freedom for the between-groups sum of squares?
- 3 (Correct answer)
- 4
- n-4
- n-1
Correct answer: 3
Between-groups degrees of freedom = k - 1, where k is the number of groups; for 4 groups, df = 3.
Question 6: A random sample of size n=36 is drawn from a population with σ=12. What is the standard error of the mean?
- 2 (Correct answer)
- 0.33
- 12
- 6
Correct answer: 2
Standard error = σ/√n = 12/√36 = 12/6 = 2.
Question 7: Which of the following best describes a conjugate prior in Bayesian analysis?
- A prior that yields a posterior in the same distributional family as the prior (Correct answer)
- A prior that is always uniform across the parameter space
- A prior chosen to maximize the likelihood function
- A prior derived from bootstrap resampling
Correct answer: A prior that yields a posterior in the same distributional family as the prior
Conjugate priors simplify Bayesian computation because the posterior has the same functional form as the prior.
MS-DS Master of Data science Statistical and Probabilistic Analysis 3
In Bayesian inference, what does the posterior distribution represent?