12. Iterated Expectations

What this lecture covers

This lecture closes out the core probability theory material of MIT 6.041 by reframing two familiar ideas, conditional expectation and conditional variance, as random variables rather than fixed numbers. It answers the question of how to compute the mean and variance of a random variable by breaking the problem into simpler conditional pieces and recombining them, an approach that becomes essential once you sum a random number of random variables.

The lecture follows the stick-breaking example from the previous class and precedes a specific application to sums with a random number of terms. After watching, you should be able to treat E[X|Y] and Var(X|Y) as random variables that are functions of Y, apply the law of iterated expectations, and use the law of total variance to split overall variability into within-group and between-group components.

Key ideas

• Conditional expectation as a random variable: E[X|Y] is a function of the random variable Y, taking the numerical value E[X|Y=y] whenever Y equals y; before the experiment, its value is unknown.
• Law of iterated expectations: E[E[X|Y]] = E[X], which is the total expectation theorem restated in random-variable notation.
• Conditional variance as a random variable: Var(X|Y) is likewise a function of Y, equal to the variance of X computed inside the conditional universe where Y is known.
• Law of total variance: Var(X) = E[Var(X|Y)] + Var(E[X|Y]), unlike expectation this needs two terms, not just E[Var(X|Y)] alone.
• Interpreting the two variance terms: E[Var(X|Y)] captures variability within each conditional scenario (such as within a section); Var(E[X|Y]) captures variability between scenarios (such as differing section averages).
• Divide-and-conquer variance calculation: splitting a continuous random variable's support into pieces and defining an auxiliary discrete random variable can make a variance calculation easier by working in each conditional piece separately.
• Sum of a random number of random variables: for i.i.d. X_1, ..., X_N independent of N, E[Y] = E[N] * E[X] and Var(Y) = E[N] * Var(X) + Var(N) * (E[X])^2, where Y = X_1 + ... + X_N.

Walkthrough

Conditional expectation reframed as a random variable (1:02)

Using the stick-breaking example (break a stick at uniform point Y, then break the remainder at a uniform point X between 0 and Y), the lecture shows that E[X|Y=y] = y/2 is a number once y is fixed, but before the experiment E[X|Y] must be treated as a random variable depending on the unknown Y.

Law of iterated expectations (7:06)

The lecture shows that the expectation of the conditional expectation, E[E[X|Y]], equals the unconditional E[X], and identifies this as the total expectation theorem in abstract notation. It applies this to recover E[X] = l/4 in the stick-breaking example.

Conditional variance and the law of total variance (11:15)

The lecture defines Var(X|Y) analogously as a random variable, then derives (via an algebraic proof) that Var(X) = E[Var(X|Y)] + Var(E[X|Y]), noting that, unlike expectation, a single term is not enough.

Quiz scores example: within-section and between-section variance (19:25)

Using a class split into two sections with different sizes and score averages, the lecture computes the overall mean via the law of iterated expectations and the overall variance via the law of total variance, interpreting the two variance terms as within-section spread and between-section spread respectively.

Divide-and-conquer variance for a continuous variable (30:40)

A piecewise-uniform density is split into two regions using an auxiliary discrete variable Y, and the lecture computes the overall variance of X by combining the conditional means and conditional variances within each region.

Sum of a random number of random variables (35:50)

Using a bookstore-shopping example (visiting a random number N of stores and spending a random amount at each), the lecture derives E[Y] = E[N]*E[X] and Var(Y) = E[N]*Var(X) + Var(N)*(E[X])^2 for Y equal to the sum of N i.i.d. terms independent of N, by conditioning on N and applying the laws developed earlier.

Before you watch

• Review the stick-breaking example and the total expectation theorem from the previous lecture on derived distributions.
• Be comfortable with the ordinary (unconditional) formulas for expectation and variance.

Check your understanding

1. Why is E[X|Y] a random variable before the experiment, but a number once Y is observed?
2. Why does the law of total variance require two terms instead of one, unlike the law of iterated expectations?
3. In the quiz scores example, what does each of the two variance terms represent?
4. Derive why E[Y] = E[N] * E[X] when Y is the sum of N i.i.d. terms independent of N.
5. Why does Var(Y) for a random sum include a term involving (E[X])^2 rather than just Var(N) * Var(X)?