VEE Mathematical Statistics
Core statistical reasoning
Mathematical statistics questions test whether you can move between a population model, a sample, and an inferential statement. Know random variables and their distributions, joint and conditional distributions, independence, expectation, variance, covariance, and moment generating functions. For estimation, distinguish a statistic from a parameter, an estimator from an estimate, and bias from variance. The mean squared error identity MSE = Var(T) + Bias(T)^2 is a frequent shortcut. Maximum likelihood and method of moments should be derived, not memorized.
Inference traps
Common traps include using n instead of n-1 for an unbiased sample variance, treating a confidence interval as a probability that the parameter lies in a fixed interval, and confusing a p-value with the probability that the null hypothesis is true. In hypothesis tests, identify the null, alternative, test statistic, rejection region, and Type I and Type II error directions before computing. For two-sided tests, double the appropriate tail probability. Check whether observations are independent and whether the variance is known.
Worked rule
If X is Bernoulli(p) and a sample has s successes in n trials, the MLE is p-hat = s/n. Its variance is p(1-p)/n, estimated by p-hat(1-p-hat)/n. A 95% large-sample interval is p-hat plus or minus 1.96 times that standard error. This illustrates the general pattern: estimate, assess uncertainty, then interpret. State model assumptions explicitly.