Stats / ML MCQ

Q1: Exponential distribution rate comparison

Three exponential pdf curves A, B, C plotted on the same axes. The curve with the highest peak at x = 0 corresponds to the largest rate parameter λ (since pdf(0) = λ), and decays fastest.

• (A) A < B < C
• (B) C < B < A
• (C) A = B = C
• (D) Cannot be determined.

Answer: B (C < B < A). Reading the typical OA plot: curve A peaks highest at 0 and drops fastest → largest λ; C peaks lowest and decays slowest → smallest λ.

Q2: Picking cards with replacement

Pick a card from a standard 52-card deck, replace it, pick again.

• P1 = P(Ace then King) = (4/52) · (4/52) = 1/169.
• P2 = P(King then Ace) = (4/52) · (4/52) = 1/169.

Answer: p1 = p2. With replacement the two draws are independent and the marginal probabilities are equal.

Q3: Skewness reduction for right-skewed data

Prepare a right-skewed dataset for linear regression. Best technique?

• (A) Square-root transformation
• (B) Replace outliers with the mode
• (C) IQR outlier filter + mean normalisation
• (D) Bivariate analysis to find correlations

Answer: A. Monotonic transforms (sqrt, log, Box-Cox) compress the right tail and reduce skewness while preserving order. Outlier replacement and bivariate analysis don't normalise the distribution.

Q4: Hypothesis testing — which statements are correct?

• (A) p-value < α ⇒ reject H0 and conclude evidence for H1.
• (B) p-value < α ⇒ reject H1 and accept H0.
• (C) p-value ≥ α ⇒ accept H0.
• (D) p-value ≥ α ⇒ fail to reject H0.

Answer: A, D. "Fail to reject H0" is not the same as "accept H0" — you simply have insufficient evidence against it.

Q5: Reducing overfitting on noisy data

The cleanest fix without inflating bias too much?

• (A) Increase model complexity
• (B) Shrink training set
• (C) L1 regularization to encourage parameter sparsity
• (D) Use plain linear regression

Answer: C. L1 (Lasso) zeroes out unimportant coefficients, reducing variance without large bias growth. (A) makes overfitting worse; (B) loses signal; (D) under-fits if the true relationship is non-linear.

Q6: Exponential distribution pmf vs pdf phrasing

The OA mislabels the exponential probability density function as a "probability mass function". The exponential distribution is continuous and has a pdf, not a pmf. Knowing this is enough to eliminate distractors that hinge on the discrete interpretation.

Answer: awareness — pdf, not pmf.