Manuscript Oai227-A@1 · Oai227-A.tex · lemma · 7f45f607

Lemma: Two sources of randomness

Stage 2: price only specified finite operations › Disorder randomness is not trajectory sampling error · lines 893–908 · proof included or linked

Oai227-A@1#lem:variance

Sample $J$ once, and conditionally on $J$ sample $N$ independent scores $Y_{1}, \dots, Y_{N}$ with conditional mean $m (J)$ . Then

Var (\frac{1}{N} k = 1 \sum^{N} Y_{k}) = \frac{1}{N} E [Var (Y_{1} ∣ J)] + Var (m (J)) .

Proof. Apply conditional independence and the law of total variance.

Displayed formulas in this passage

unlabeled · line 898
$Var (\frac{1}{N} k = 1 \sum^{N} Y_{k}) = \frac{1}{N} E [Var (Y_{1} ∣ J)] + Var (m (J)) .$