Total expectation with a nonempty null block

Goal and scope

Audit the partition and compute E[X] by two routes.

Finite law; no conditional division on a null cell.

Givens and symbols

• work

• G1 — Omega={a,b,c,d,e} has masses(1/8,3/8,1/4,1/4,0), X=(0,2,4,4,99); B1={a,b}, B2={c,d}, B0={e}, and G is their generated sigma-algebra.

• G — partition-generated sigma-algebra
• Y — conditional-expectation version
• k — null-block version value

Rules and sources

Some rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged.

• mko_definition · cited — Let G be the sigma-algebra generated by a complete partition. The tower identity is E(E(X|G))=E(X). Cells are pairwise disjoint and cover the sample space. Positive-mass cells contribute P(B_i)E[X|B_i]. Conditional expectation is determined only almost surely: a null-cell quotient is undefined, but any integrable version there has zero weighted contribution. Do not divide by zero. No independence is required, and this is not Bayes formula. · mko-law-of-total-expectation
• work · prerequisite — On a positive-mass block, a conditional mean is its probability-weighted sum divided by the block mass; on a null block, any finite version value may be chosen; finite sums may be regrouped.

Main derivation

\{B_1,B_2,B_0\}

B_i\cap B_j=\varnothing\ (i\ne j),\quad\bigcup_iB_i=\Omega

P(B_1),P(B_2),P(B_0)

(P(B_1),P(B_2),P(B_0))=(1/2,1/2,0)

E[X\mid B_1]=\frac{0/8+2(3/8)}{1/2}

E[X\mid B_1]=3/2

E[X\mid B_2]=\frac{4/4+4/4}{1/2}

E[X\mid B_2]=4

P(B_0)=0

Y=E[X\mid G]=(3/2,3/2,4,4,k),\quad k\in\mathbb R

E[Y]=\tfrac12\tfrac32+\tfrac12 4+0k

E[Y]=11/4

E[X]=0/8+2(3/8)+4/4+4/4+99(0)

E[X]=11/4

\sum_{i:P(B_i)>0}P(B_i)\frac{\sum_{\omega\in B_i}p_\omega X(\omega)}{P(B_i)}

\sum_{\omega\in\Omega}p_\omega X(\omega)=E[X]

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S07, work — The original weighted sum on null atoms is zero as well.
• S07 · premise_audit · G1, S06, work — The term99 times zero does not force a conditional version value 99.
• S06 · premise_audit · G1, S05, work — Every finite k gives the same outer expectation.
• S05 · premise_audit · G1, S04, work — Versions may differ at e only on a null set.
• S04 · premise_audit · G1, S03, work — The numerator 2 must be divided by 1/2.
• S03 · premise_audit · G1, S02, work — The numerator 3/4 is not the conditional mean.
• S02 · premise_audit · G1, S01, work — B0 is{e}, not the empty set.
• S01 · premise_audit · G1, work — The zero-mass atom e must not disappear from the carrier.

Error controls and negative controls

Null block

Compute cell masses before deciding whether division is allowed.

Two-route audit

Compare the direct sum and outer conditional sum at 11/4.

• E[X|B0]=0/0=0. → reject — The quotient is undefined; one may choose a version, not define it by division.
• The partition blocks must be independent. → reject — Total expectation needs disjointness and exhaustiveness, not independence.
• Omitting b leaves the same partition. → reject — b has positive mass 3/8 and coverage would fail.

Result and limitations

E(E[X|G])=E[X]=11/4, independently of the null-block version value.

Verification and source binding

2 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 24bfde5c0b3950c83c7e98fb04fbc75973b2b5a7422e8b9f5843de4153da3a73
• original source SHA-256: 743b092227a8a2ca350616cc5f1b13f4b757db1ac765c638dd10d4dd33fe489d
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

Same-source full JSON (formulas, ASTs, controls and revision log)