Exam PMultivariate probabilityFree to read

Conditional expectation and the variance decomposition

Condition on what you do not know, then average it out. The double expectation and the conditional variance formula handle every compound model on the syllabus.

The formulas

Double expectation

$$E[X]=E[E[X\mid Y]]$$

Conditional variance formula

$$\mathrm{Var}(X)=E[\mathrm{Var}(X\mid Y)]+\mathrm{Var}(E[X\mid Y])$$

Compound mean

$$E[S]=E[N]E[X]$$

Compound variance

$$\mathrm{Var}(S)=E[N]\mathrm{Var}(X)+\mathrm{Var}(N)E[X{]}^2$$

Where it comes from

1. $E[X\mid Y]$ is a random variable — a function of $Y$ — and averaging it over $Y$ recovers $E[X]$.
2. Splitting $\mathrm{Var}(X)$ into the average within-group variance and the variance of the group means gives the decomposition.
3. Applying both to a random sum $S=X_1+\cdots +X_N$ produces the compound formulas, with the second term carrying the uncertainty in the COUNT.

Worked example

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

• 'Mean of the variance plus variance of the mean' — say it out loud and the formula writes itself.
• For a Poisson frequency the two terms collapse to λ·E[X²].

Traps

• Using Var(N)·Var(X) somewhere in the compound variance — neither term has that form.
• Squaring E[X] in the wrong term: the SECOND term carries E[X]².

Related

• Joint, marginal and conditional distributions
• Poisson distribution
• Expectation, variance and moments

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