Conditional expectation

• Conditional Mean and Variance of an Exponential Random Variable
  For an exponential random variable conditioned to exceed a threshold, memorylessness gives a shifted exponential: mean threshold plus 1/rate and variance 1/rate squared.
• 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.
• How to Prove the Law of Total Expectation
  The law of total expectation says E[Y] = E[E[Y|X]]. Learn its proof from conditional expectation's definition, with discrete, continuous, and worked numerical examples.
• Law of Iterated Expectations: A Worked Example
  The law of iterated expectations says to average a conditional mean over the variable you conditioned on; a weighted two-bag example shows each step.
• What Is the Tower Property of Conditional Expectation?
  The tower property says that conditioning first on finer information and then on coarser information is the same as conditioning directly on the coarser information.