Probability theory

• How Does Expected Value Change with the Probability Measure?
  Expected value depends on the probability measure: integrate the same random variable under each measure, or use a density to convert between them.
• How to Prove the Main Properties of Conditional Expectation
  Prove linearity, positivity, the tower property, and other conditional expectation identities from its defining integral property.
• What Is the MGF of a Uniform Distribution?
  For a continuous uniform random variable on [a,b], the moment-generating function is (e^{bt}-e^{at})/[t(b-a)], with value 1 at t=0.