Nuprl Lemma : expectation-rv-add
∀[p:FinProbSpace]. ∀[n:ℕ]. ∀[X,Y:RandomVariable(p;n)]. (E(n;X + Y) = (E(n;X) + E(n;Y)) ∈ ℚ)
Proof
Definitions occuring in Statement :
expectation: E(n;F),
rv-add: X + Y,
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
qadd: r + s,
rationals: ℚ,
nat: ℕ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
prop: ℙ,
squash: ↓T,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
rv-scale: q*X,
rv-add: X + Y,
nat: ℕ,
rev_implies: P ⇐ Q
Lemmas referenced :
expectation-linear,
int-subtype-rationals,
equal_wf,
squash_wf,
true_wf,
rationals_wf,
expectation_wf,
iff_weakening_equal,
qadd_wf,
qmul_wf,
qmul_ident,
random-variable_wf,
nat_wf,
finite-prob-space_wf,
int_seg_wf,
length_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
applyEquality,
sqequalRule,
because_Cache,
hyp_replacement,
equalitySymmetry,
lambdaEquality,
imageElimination,
equalityTransitivity,
universeEquality,
imageMemberEquality,
baseClosed,
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 + Y) = (E(n;X) + E(n;Y)))
Date html generated:
2018_05_22-AM-00_34_47
Last ObjectModification:
2017_07_26-PM-06_59_55
Theory : randomness
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