Peter Kempthorne continues MIT's 18.642, Topics in Mathematics with Applications in Finance, with a lecture spanning probability theory and the start of stochastic processes. He covers Principal Components Analysis, the technique of shifting and rotating coordinates in a multidimensional random vector to find orthogonal directions of maximum variability, useful for simplifying covariance structures in financial data. The lecture also revisits the Central Limit Theorem, discusses utility optimization in asset pricing, and introduces martingales as a foundational stochastic process for modeling markets. Kempthorne works through the mathematical formulations on the board, connecting abstract probability tools to their application in quantitative finance. This is lecture six of the course, building on prior sessions' probability groundwork before moving into the stochastic process material that underlies option pricing and risk models later in the term.