Nuprl Lemma : expectation-rv-scale
∀[p:FinProbSpace]. ∀[n:ℕ]. ∀[X:RandomVariable(p;n)]. ∀[q:ℚ]. (E(n;q*X) = (q * E(n;X)) ∈ ℚ)
Proof
Definitions occuring in Statement :
expectation: E(n;F),
rv-scale: q*X,
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
qmul: 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,
rv-const: a,
rv-scale: q*X,
rv-add: X + Y,
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
qmul: r * s,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
ifthenelse: if b then t else f fi ,
btrue: tt,
nat: ℕ,
and: P ∧ Q,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
expectation-linear,
rv-const_wf,
int-subtype-rationals,
equal_wf,
squash_wf,
true_wf,
expectation_wf,
mon_ident_q,
qmul_wf,
int_seg_wf,
length_wf,
rationals_wf,
random-variable_wf,
nat_wf,
finite-prob-space_wf,
qadd_wf,
qmul_zero_qrng,
qmul_comm_qrng,
qadd_comm_q,
iff_weakening_equal
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,
functionExtensionality,
functionEquality,
setElimination,
rename,
productElimination,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[p:FinProbSpace]. \mforall{}[n:\mBbbN{}]. \mforall{}[X:RandomVariable(p;n)]. \mforall{}[q:\mBbbQ{}]. (E(n;q*X) = (q * E(n;X)))
Date html generated:
2018_05_22-AM-00_34_49
Last ObjectModification:
2017_07_26-PM-06_59_56
Theory : randomness
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