Section 12.1

The Tower Property

Two-stage experiments and random sums: E[E[X|Y]], compound Poisson counts, the law of total variance, mixtures.

Toolkit

Many two-stage problems become tractable by conditioning on the “outer” randomness and using the tower property:

\mathbb{E}[X] = \mathbb{E}\big[\mathbb{E}[X\mid Y]\big],

valid whenever the inner conditional expectation is well-defined; iterate it for nested random sums or random parameters. For variances, decompose via the law of total variance:

\operatorname{Var}(X) = \mathbb{E}\big[\operatorname{Var}(X\mid Y)\big] + \operatorname{Var}\big(\mathbb{E}[X\mid Y]\big).

For a random sum $S = \sum_{i=1}^N X_i$ with $N$ independent of the i.i.d. $X_i$ ,

\mathbb{E}[S] = \mathbb{E}[N]\,\mathbb{E}[X], \qquad \operatorname{Var}(S) = \mathbb{E}[N]\operatorname{Var}(X) + \operatorname{Var}(N)\,\mathbb{E}[X]^2.

Binomial thinning of a Poisson count with independent keep-probability $p$ yields a $\mathrm{Poisson}(p\lambda)$ count; mixtures (random parameters of a known distribution) are handled the same way, conditioning on the parameter.

Problems