MS-DS Master of Data science Statistical and Probabilistic Analysis 3

Question 1

In Bayesian inference, what does the posterior distribution represent?

Accepted Answer

Updated belief about a parameter after observing data

Answer

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?

Accepted Answer

Student's t-distribution

Answer

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?

Accepted Answer

To reduce skewness and stabilize variance

Answer

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?

Accepted Answer

Normal distribution

Answer

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?

Accepted Answer

3

Answer

Between-groups degrees of freedom = k - 1, where k is the number of groups; for 4 groups, df = 3.

MS-DS Master of Data science Practice Test