Nuprl Lemma : expectation-rv-add-squared
∀[p:FinProbSpace]. ∀[n:ℕ]. ∀[X,Y:RandomVariable(p;n)].
(E(n;(x.x * x) o X + Y) = ((E(n;(x.x * x) o X) + (2 * E(n;X * Y))) + E(n;(x.x * x) o Y)) ∈ ℚ)
Proof
Definitions occuring in Statement :
rv-compose: (x.F[x]) o X,
expectation: E(n;F),
rv-mul: X * Y,
rv-add: X + Y,
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
qmul: r * s,
qadd: r + s,
rationals: ℚ,
nat: ℕ,
uall: ∀[x:A]. B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
random-variable: RandomVariable(p;n),
p-outcome: Outcome,
rv-compose: (x.F[x]) o X,
rv-mul: X * Y,
rv-scale: q*X,
rv-add: X + Y,
nat: ℕ,
qadd: r + s,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
ifthenelse: if b then t else f fi ,
btrue: tt
Lemmas referenced :
random-variable_wf,
nat_wf,
finite-prob-space_wf,
rationals_wf,
expectation_wf,
rv-compose_wf,
rv-add_wf,
qmul_wf,
rv-mul_wf,
int-subtype-rationals,
rv-scale_wf,
squash_wf,
true_wf,
equal_wf,
qadd_wf,
expectation-rv-scale,
iff_weakening_equal,
expectation-rv-add,
int_seg_wf,
p-outcome_wf,
qmul_over_plus_qrng,
qmul_comm_qrng,
mon_assoc_q,
qadd_comm_q,
q_distrib,
qmul_ident,
qmul_assoc_qrng
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
lambdaEquality,
natural_numberEquality,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
functionExtensionality,
functionEquality,
setElimination,
rename
Latex:
\mforall{}[p:FinProbSpace]. \mforall{}[n:\mBbbN{}]. \mforall{}[X,Y:RandomVariable(p;n)].
(E(n;(x.x * x) o X + Y) = ((E(n;(x.x * x) o X) + (2 * E(n;X * Y))) + E(n;(x.x * x) o Y)))
Date html generated:
2018_05_22-AM-00_35_45
Last ObjectModification:
2017_07_26-PM-07_00_15
Theory : randomness
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