Nuprl Lemma : rv-identically-distributed_wf
∀[p:FinProbSpace]. ∀[f:ℕ ⟶ ℕ]. ∀[X:n:ℕ ⟶ RandomVariable(p;f[n])]. (rv-identically-distributed(p;n.f[n];n.X[n]) ∈ ℙ)
Proof
Definitions occuring in Statement :
rv-identically-distributed: rv-identically-distributed(p;n.f[n];i.X[i]),
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
rv-identically-distributed: rv-identically-distributed(p;n.f[n];i.X[i]),
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s]
Lemmas referenced :
all_wf,
nat_wf,
equal_wf,
rationals_wf,
expectation_wf,
rv-compose_wf,
qmul_wf,
random-variable_wf,
finite-prob-space_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
lambdaEquality,
because_Cache,
productEquality,
hypothesisEquality,
applyEquality,
functionExtensionality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
isect_memberEquality
Latex:
\mforall{}[p:FinProbSpace]. \mforall{}[f:\mBbbN{} {}\mrightarrow{} \mBbbN{}]. \mforall{}[X:n:\mBbbN{} {}\mrightarrow{} RandomVariable(p;f[n])].
(rv-identically-distributed(p;n.f[n];n.X[n]) \mmember{} \mBbbP{})
Date html generated:
2018_05_22-AM-00_37_47
Last ObjectModification:
2017_07_26-PM-07_00_41
Theory : randomness
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