What is an undergraduate math stat course?
An undergraduate mathematical statistics course that has a probability course as a prerequisite should have covered the use of standard statistical inference techniques such as confidence intervals and hypothesis tests and the mathematical foundations of such techniques such as maximum likelihood estimation and likelihood ratio tests and comparison of estimators and comparison of hypothesis tests.
A course including these mathematical foundations will have required the use of probability theory to derive these results including familiarity with probability density (mass) functions, cumulative distribution functions, and moment generating functions. Results from probability theory will have been used to derive properties of the sampling distributions of various statistics, including , at a minimum, various statistics based on samples from the uniform, binomial, exponential, and normal (Gaussian) distributions, and resulting in various distributions including t, F, and chi-squared distributions.
Preferably students will have also used probability techniques to derive properties of other distributions such as gamma and beta, requiring skill at handling two parameters in a variety of situations.