Discrete Stochastic Processes
Conditional Expectation for Dummies
Seeing the first die rules out 30 of 36 worlds and moves the expected sum away from 7, yet averaged over every face you might see, the forecast is still 7.
What to remember
- Conditioning on an event: delete the ruled-out worlds, rescale the rest, find the new balance point. Two dice with the first showing 5: E[X | D₁ = 5] = 8.5.
- E[X | Y] is a random variable, a function of Y: E[sum | D₁] = D₁ + 3.5 and E[D₁ | sum] = sum/2.
- Tower property: E[E[X | Y]] = E[X]. News moves the forecast, but not on average; with nested information the coarser one wins.
- Five working rules: linearity, taking out what is known, iterated conditioning, independence, and Jensen. For a convex payoff the average payoff is at least the payoff at the average.
- Law of total variance: Var(X) = E[Var(X | Y)] + Var(E[X | Y]). The first die removes half of the variance of the sum (35/12 of 35/6).
- In two-stage problems, condition on the first stage, then average. Never plug E[N] into a nonlinear function.
Read the lessonConditional Expectation and the Tower Property, with a checkpoint at the end.