20 interleaved questions drawn from every lesson in this phase. Interleaving mixes topics on purpose - that difficulty is what builds durable, transferable understanding. Aim for 70%+ before advancing; below that, revisit the flagged lessons.
Q1 EasyAn estimator has bias 0.2 and variance 0.09. Its mean-squared error is:
Q2 MediumThe normal equations \(X^\top X\hat\beta=X^\top y\) express the condition that:
The fitted values equal the data
The residual vector is orthogonal to the column space of \(X\)
The coefficients are all positive
The design matrix is orthonormal
Q3 MediumTraining error is a biased estimate of test error because:
It uses too few data points
The same fit that lowers it also fits the noise in the training points
It ignores the intercept
It is computed with the wrong loss
Q4 MediumThe ridge estimator \((X^\top X+\lambda I)^{-1}X^\top y\) is always well-defined even when OLS is not because:
It ignores collinear columns
Adding \(\lambda I\) makes the matrix positive definite hence invertible
It uses the pseudo-inverse
It drops the intercept
Q5 MediumIn logistic regression, a coefficient \(\beta_j\) is best interpreted as:
The change in probability per unit \(x_j\)
The change in log-odds per unit \(x_j\)
The class prior
The residual variance
Q6 MediumBagging improves a single deep tree primarily by:
Reducing bias
Reducing variance by averaging many high-variance trees
Making the tree deeper
Removing the need for cross-validation
Q7 MediumYou standardize all features using the mean and standard deviation of the entire dataset, then split into train/test. This causes:
No problem if you use the same scaler
Data leakage: test statistics influenced the training pipeline
Only a numerical rounding issue
Underfitting
Q8 MediumThe maximum-likelihood variance \(\hat\sigma^2=\tfrac1n\sum(x_i-\bar x)^2\) is biased because:
It divides by \(n-1\)
It reuses \(\bar x\) in place of the true mean, removing one degree of freedom
The data are not Gaussian
Likelihood maximization always biases downward
Q9 MediumThe Gauss-Markov theorem guarantees OLS is best among:
All estimators
All unbiased estimators
All linear unbiased estimators
All linear estimators
Q10 MediumCompared with 10-fold CV, leave-one-out CV generally has:
Higher bias, lower variance
Lower bias, higher variance
Lower bias and lower variance
Identical behavior
Q11 MediumWhy does the lasso produce exactly-zero coefficients while ridge does not?
Lasso uses a smaller \(\lambda\)
The \(\ell_1\) constraint region has corners on the axes where the solution tends to land
Lasso is not convex
Ridge has no closed form
Q12 MediumWhy does logistic regression require iterative optimization?
The likelihood is non-convex
The score equation \(X^\top(y-p)=0\) is nonlinear in \(\beta\) because \(p=\sigma(X\beta)\)
There are too many parameters
The data must be standardized first
Q13 MediumBoosting differs from bagging in that it:
Fits trees in parallel on bootstrap samples
Fits shallow trees sequentially, each to the current residuals/gradient
Always uses deep trees
Cannot overfit
Q14 MediumTrading a signal that is computed from today's closing price, executed at today's close, exhibits:
Survivorship bias
Look-ahead bias
Selection bias
No bias
Q15 MediumIn the prediction-error decomposition, the \(\sigma^2\) term represents:
Model bias you can remove with a bigger model
Estimation variance that shrinks with more data
Irreducible noise that no model can eliminate
A numerical rounding error
Q16 MediumYou want the most accurate forecast of tomorrow's return, not to interpret coefficients. Which matters most?
Unbiasedness of each coefficient
Statistical significance of each factor
Cross-validated out-of-sample error
A high in-sample \(R^2\)
Q17 MediumWhich practice most directly causes an optimistic, leaked CV score in finance?
Using K=10 instead of K=5
Standardizing features using statistics from the entire dataset before splitting
Using squared-error loss
Reporting the CV standard error
Q18 MediumAs the penalty \(\lambda\) increases from 0, the bias and variance of the estimator:
Both increase
Bias decreases, variance increases
Bias increases, variance decreases
Both decrease
Q19 MediumLDA yields a linear boundary because:
It uses least squares
The shared covariance makes the quadratic \(x^\top\Sigma^{-1}x\) terms cancel between classes
The priors are equal
It ignores the covariance
Q20 MediumIn a soft-margin SVM, the decision boundary is determined by:
All training points equally
Only the support vectors (points on or inside the margin)
The class means
The most distant points