Widgets and Crates

What this lecture covers

This is a worked recitation problem applying the law of iterated expectations and the law of total variance to a random sum: a crate holds a random number N of boxes, each box holds a random number of widgets, and the problem finds the expectation and variance of the total number of widgets T. It is similar in structure to the stick-breaking problem covered earlier, but here the randomness is in how many terms are summed rather than in a nested interval.

After watching, you should be able to recognize when a quantity is a sum of a random number of independent, identically distributed random variables, and compute its mean and variance by conditioning on the number of terms.

Key ideas

• Setup: T = X1 + X2 + ... + XN, where N (number of boxes) and each Xi (widgets per box) are independent, with E[Xi] = 10, Var(Xi) = 16, E[N] = 10, Var(N) = 16.
• Condition on N to simplify: if N were a fixed number n, E[sum] = n * E[Xi] and Var(sum) = n * Var(Xi) follow directly from linearity of expectation and independence.
• Law of iterated expectations gives the mean: E[T] = E[N] * E[Xi] = 10 * 10 = 100.
• Law of total variance gives the variance: Var(T) = E[Var(T|N)] + Var(E[T|N]), where E[Var(T|N)] = E[N] * Var(Xi) = 160 and Var(E[T|N]) = (E[Xi])^2 * Var(N) = 1600, giving Var(T) = 1760.
• Independence is essential: the simplification relies on the widget counts per box not depending on how many boxes there are.
• Hierarchical conditioning: problems with a natural hierarchy (widgets inside boxes inside crates) are naturally solved by conditioning at each level and building up.

Before you watch

• Watch the main lecture on iterated expectations and, ideally, the stick-breaking recitation problem, since this problem reuses the same two laws in a similar divide-and-conquer style.
• Be comfortable with linearity of expectation and the variance formula for a sum of independent random variables.

Check your understanding

1. Why does conditioning on the number of boxes N make the problem tractable?
2. Derive E[T] = 100 using the law of iterated expectations.
3. Walk through both terms of the law of total variance calculation and explain what each one represents.
4. Why is independence between N and the Xi's necessary for these formulas to hold?