GATE ST study plan

Derive the MGF, mean, and variance of the binomial, Poisson, geometric, negative binomial, and hypergeometric distributions without notes.

Compute the MGF of the normal, exponential, gamma, and uniform distributions and use it to identify the distribution of a sum of independent variables.

Classify a stated random variable as discrete or continuous and select the correct pmf or pdf, including indicator-function notation for support.

Solve 15 distribution-identification problems under a 25-minute timer and log every error with its cause.

Apply the convolution method and the MGF method to find the distribution of sums of independent random variables.

Compute joint, marginal, and conditional densities and verify independence from the factorization of the joint density.

Use the Jacobian transformation technique to derive the distribution of functions of two random variables.

Compute covariance and correlation from a joint density and construct a counterexample where zero correlation does not imply independence.

State and prove the weak law of large numbers via Chebyshev's inequality and apply it to bound tail probabilities numerically.

State the central limit theorem and use it to approximate binomial and Poisson probabilities with continuity correction.

Apply Slutsky's theorem and the delta method to find limiting distributions of transformed sequences.

Distinguish convergence in probability, in distribution, and almost surely using concrete counterexamples.

Derive the sampling distributions of the sample mean and sample variance for normal samples and prove independence of the two.

Derive the t, chi-square, and F distributions from normal, chi-square, and independent chi-square building blocks.

Construct confidence intervals for a mean, a difference of means, a variance, and a proportion, stating every assumption used.

Identify the correct pivot quantity for a given parametric setting and invert it to obtain the interval.

Derive the maximum likelihood estimator for at least six models, including cases where the support depends on the parameter.

Compute the Fisher information and the Cramer-Rao lower bound and check whether an estimator attains it.

Apply the method of moments and compare the resulting estimator with the MLE in bias and variance.

Evaluate the small-sample bias of the MLE for the variance of a normal distribution and state the unbiased correction.

Construct the Neyman-Pearson most powerful test for simple versus simple hypotheses and compute its power function.

Apply the Karlin-Rubin theorem to find uniformly most powerful tests for one-sided hypotheses in monotone likelihood ratio families.

Derive the likelihood ratio test for a composite null and state its asymptotic chi-square distribution with the correct degrees of freedom.

Compute the probability of a type II error and the sample size needed for a specified power in a normal-mean test.

Fit simple and multiple linear regression models by least squares and derive the normal equations and the hat matrix.

Interpret the coefficient of determination, adjusted R-squared, and the standard errors of the regression coefficients.

Perform one-way analysis of variance, construct the ANOVA table, and carry out the F-test for equality of means.

Diagnose violations of regression assumptions from residual patterns and state the appropriate remedy.

Solve three full-length timed mock papers under exam conditions and record raw scores and time per section.

Re-derive every formula that was missed in the mocks from first principles without consulting notes.

Reduce the personal error log to the five recurring failure modes and write a targeted fix for each.

Revise the distribution cheat sheet and the estimator and test summary into a single final one-page reference.