Nuprl Lemma : expectation-constant
∀[p:FinProbSpace]. ∀[a:ℚ]. ∀[n:ℕ]. ∀[X:RandomVariable(p;n)]. E(n;X) = a ∈ ℚ supposing ∀s:ℕn ─→ Outcome. ((X s) = a ∈ ℚ)
Proof
Definitions occuring in Statement :
expectation: E(n;F),
random-variable: RandomVariable(p;n),
p-outcome: Outcome,
finite-prob-space: FinProbSpace,
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
function: x:A ─→ B[x],
natural_number: $n,
equal: s = t ∈ T,
rationals: ℚ
Lemmas :
all_wf,
int_seg_wf,
p-outcome_wf,
equal_wf,
rationals_wf,
random-variable_wf,
nat_wf,
finite-prob-space_wf,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
le_wf,
decidable__le,
subtract_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
null-seq_wf,
iff_weakening_equal,
length_wf,
eq_int_wf,
bool_wf,
equal-wf-base,
int_subtype_base,
assert_wf,
bnot_wf,
not_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
ws-constant,
expectation_wf,
not-le-2,
rv-shift_wf,
cons-seq_wf,
trivial-int-eq1,
subtype_rel_self
\mforall{}[p:FinProbSpace]. \mforall{}[a:\mBbbQ{}]. \mforall{}[n:\mBbbN{}]. \mforall{}[X:RandomVariable(p;n)].
E(n;X) = a supposing \mforall{}s:\mBbbN{}n {}\mrightarrow{} Outcome. ((X s) = a)
Date html generated:
2015_07_17-AM-07_58_55
Last ObjectModification:
2015_02_03-PM-09_44_27
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