The Variance in the Stick Breaking Problem

What this lecture covers

This is a worked recitation problem applying two tools from the main lecture on iterated expectations: the law of iterated expectations and the law of total variance. It answers a concrete question: if a stick of length l is broken uniformly at random, and the remaining left piece is broken uniformly at random again, what are the expectation and variance of the final piece's length?

The problem shows how conditioning on the first break turns a hard two-stage calculation into two easy conditional calculations combined with the two laws. After watching, you should be able to identify what to condition on to simplify a nested random process, and apply both laws to compute a mean and a variance in stages.

Key ideas

• Condition on the useful intermediate variable: letting Y be the length after the first break makes the second break simple to analyze, since X is uniform between 0 and Y once Y is known.
• Conditional expectation and variance become simple formulas: given Y = y, E[X|Y=y] = y/2 and Var(X|Y=y) = y^2/12, the standard formulas for a variable uniform on [0, y].
• Law of iterated expectations applied: E[X] = E[E[X|Y]] = E[Y/2] = E[Y]/2 = l/4, since Y itself is uniform on [0, l].
• Law of total variance applied: Var(X) = E[Var(X|Y)] + Var(E[X|Y]), computed here as l^2/36 + l^2/48 = 7l^2/144.
• Choosing the conditioning variable matters: not every choice of variable to condition on simplifies the problem; picking a useful one is a skill built through practice.

Before you watch

• Watch the main lecture on iterated expectations first, since this problem directly applies the law of iterated expectations and the law of total variance introduced there.
• Know the mean and variance formulas for a uniform random variable.

Check your understanding

1. Why does conditioning on the length after the first break make the second stage of the problem easy to analyze?
2. Derive E[X] = l/4 using the law of iterated expectations.
3. Why does the law of total variance need two separate terms, and what does each term represent in this problem?
4. What criteria make a variable a good choice to condition on when simplifying a multi-stage random process?