Nuprl Lemma : expectation-linear
∀[p:FinProbSpace]. ∀[n:ℕ]. ∀[X,Y:RandomVariable(p;n)]. ∀[a,b:ℚ]. (E(n;a*X + b*Y) = ((a * E(n;X)) + (b * E(n;Y))) ∈ ℚ)
Proof
Definitions occuring in Statement :
expectation: E(n;F),
rv-scale: q*X,
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],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
expectation: E(n;F),
ycomb: Y,
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
rv-scale: q*X,
rv-add: X + Y,
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
p-outcome: Outcome,
int_seg: {i..j-},
lelt: i ≤ j < k,
sq_stable: SqStable(P),
squash: ↓T,
nat_plus: ℕ+,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
rationals_wf,
random-variable_wf,
qadd_wf,
qmul_wf,
int_seg_wf,
length_wf,
null-seq_wf,
le_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
finite-prob-space_wf,
eq_int_wf,
bool_wf,
equal-wf-base,
int_subtype_base,
assert_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
bnot_wf,
not_wf,
decidable__equal_int,
sq_stable__and,
sq_stable__le,
sq_stable__less_than,
member-less_than,
squash_wf,
weighted-sum_wf2,
expectation_wf,
rv-shift_wf,
equal_wf,
rv-add_wf,
rv-scale_wf,
ws-linear,
iff_weakening_equal,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
true_wf,
p-outcome_wf,
rv-shift-linear
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
applyEquality,
functionExtensionality,
functionEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
unionElimination,
baseApply,
closedConclusion,
baseClosed,
productElimination,
imageMemberEquality,
imageElimination,
equalityElimination,
impliesFunctionality,
universeEquality
Latex:
\mforall{}[p:FinProbSpace]. \mforall{}[n:\mBbbN{}]. \mforall{}[X,Y:RandomVariable(p;n)]. \mforall{}[a,b:\mBbbQ{}].
(E(n;a*X + b*Y) = ((a * E(n;X)) + (b * E(n;Y))))
Date html generated:
2018_05_22-AM-00_34_40
Last ObjectModification:
2017_07_26-PM-06_59_53
Theory : randomness
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