Every such lecture begins with a quiet but absolute premise: before inference comes probability. But not the playful probability of dice and cards. This is probability as a branch of measure theory. The professor will draw the holy trinity on the board: the sample space ( \Omega ), the sigma-algebra ( \mathcalF ), and the probability measure ( P ). A random variable is not merely a number; it is a measurable function from this abstract space to the real line.

We make few or no assumptions about the underlying distribution family. The structure of the model is not fixed a priori.

A is a function that maps outcomes of a random experiment to real numbers.

( = 1 - \beta = P(\textReject H_0 \mid H_a \text true) ).