Part IV — Expectation and Moments¶
Part III built laws; this part compresses them. A moment is what you get by averaging a power of a random variable, and averaging is the operation that trades an entire distribution for a handful of numbers. The compression is what makes portfolio arithmetic possible, because a few numbers combine where whole distributions do not — and it is dangerous for exactly the same reason, because the combination rules are silent about which of the numbers exist and which do not.
The dependencies run in file order, with one exception worth knowing: Conditional Expectation owns both \(\mathbb{E}[X\mid Y=y]\) and \(\mathbb{E}[X\mid Y]\), and Law of Total Expectation and Law of Total Variance are both restatements of the object it builds, so 06 reads before either — even though the event-partition version those two generalize was proved back in Law of Total Probability and is not reproved here. Where this part stops is worth stating too: no named family's moments are tabulated, which is Part V; nothing is a vector and no matrix appears, because random vectors and covariance matrices are Part VI; no limit in \(n\) is ever taken, so Markov's and Chebyshev's inequalities are here but the law of large numbers is Part VII; and no property of an estimator is named, which is Part X and Part XI.
Topics¶
| Topic | Focus |
|---|---|
| Expected Value | The definition and the condition it requires, a distribution with no mean at all, linearity without assumptions, the law of the unconscious statistician, Jensen, Markov, and the moment nobody can measure |
| Variance | The second central moment, why the computational formula is unsafe, standardization, Chebyshev's distribution-free bound, and the variance of a sum that the \(\sqrt{T}\) rule assumes away |
| Higher-Order Moments | Skewness and kurtosis with the \(-3\) derived, the moments a two-regime mixture manufactures, the order up to which a heavy-tailed law has any moments, and what a sample kurtosis measures when there is nothing to measure |
| Covariance | The definition and its shortcut, bilinearity, the portfolio double sum where the cross terms outnumber the diagonal, zero covariance under total dependence, and positive semi-definiteness stated once |
| Correlation | Standardizing a covariance, the bound as Cauchy–Schwarz, correlation as a cosine, what it cannot see, Spearman's rank correlation as a functional of the copula alone, and the \(1/\rho\) ceiling on diversification |
| Conditional Expectation | The number and the random variable, the properties that follow, orthogonality, and the best-predictor theorem that every forecasting model in this book is written against |
| Law of Total Expectation | The tower property, iterating across nested information, Wald's identity for a random number of terms, and the exact accounting for a number reported because it survived a filter |
| Law of Total Variance | The within-and-between decomposition as a Pythagorean identity, explained variance as a ratio rather than a convention, and the variance of a sum of a random number of terms |