GATE DA study plan

Derive and apply Bayes' theorem and conditional-expectation identities to at least 20 mixed discrete/continuous problems, writing each solution out in full.

Compute expectation and variance for the standard distributions (uniform, binomial, Poisson, exponential, normal, geometric) from their definitions rather than memorized formulas.

Prove linearity of expectation and use indicator variables to solve at least 8 'expected number of' counting problems.

State and apply the central limit theorem and the law of large numbers to decide when a normal approximation is justified.

Construct maximum-likelihood estimators for at least 6 parametric models and verify unbiasedness where it holds.

Differentiate between bias, variance, and mean squared error, and compute the bias-variance decomposition for a simple estimator.

Apply hypothesis testing logic (null/alternative, p-value, Type I and Type II error) to at least 10 scenario questions without relying on a reference table beyond the normal and t distributions.

Derive the OLS solution for simple and multiple linear regression in matrix form and interpret each coefficient.

Reduce at least 15 matrices to row-echelon form and extract rank, null space, and column space from each.

Compute eigenvalues and eigenvectors by hand for 2x2 and 3x3 matrices and diagonalize where possible.

Classify quadratic forms as positive definite, negative definite, or indefinite using leading principal minors.

Apply singular value decomposition to a small matrix and interpret the singular values as variance along principal directions.

Compute gradients, Jacobians, and Hessians of at least 12 multivariable functions, including composite and vector-valued ones.

Derive the gradient of the least-squares loss and of the logistic loss from first principles.

Apply the chain rule through a two-layer neural network by hand on a small numeric example.

Use Lagrange multipliers to solve 5 constrained optimization problems.

Implement linear regression, logistic regression, and k-nearest neighbors from scratch in Python using only NumPy, with gradient-checked training loops.

Derive and implement the closed-form solution for ridge regression and compare its test error to ordinary least squares on a synthetic dataset.

Explain the bias-variance tradeoff in terms of model complexity and demonstrate it empirically by varying polynomial degree.

Implement k-fold cross-validation and use it to select a hyperparameter.

Derive the decision-tree split criterion for both Gini impurity and entropy, and hand-build a depth-2 tree on a 10-row dataset.

Implement a random forest wrapper over your own tree code and report out-of-bag error.

Derive the support vector machine margin maximization problem and explain the role of the kernel trick without re-deriving the dual in full.

Implement a single-layer perceptron and a small multilayer perceptron, and identify when the perceptron fails to converge.

Implement k-means and hierarchical clustering, and evaluate both with silhouette score on the same data.

Derive the expectation-maximization updates for a two-component Gaussian mixture and implement them.

Apply principal component analysis via eigen-decomposition and via SVD, and confirm the two agree numerically.

Distinguish supervised from unsupervised evaluation metrics and justify the choice for each algorithm above.

Solve two full-length timed mock papers under exam conditions and log every error by topic.

Re-derive from memory the 15 formulas you missed most often during the mock papers.

Revisit your weakest four subtopics from the error log and solve 10 fresh problems in each.

Assemble a one-page final revision sheet limited to formulas and decision rules you actually got wrong at least once.