Nuprl Lemma : expectation-qsum
∀[k:ℕ]. ∀[p:FinProbSpace]. ∀[n:ℕ]. ∀[X:ℕk ─→ RandomVariable(p;n)].
(E(n;λs.Σ0 ≤ i < k. X i s) = Σ0 ≤ i < k. E(n;X i) ∈ ℚ)
Proof
Definitions occuring in Statement :
expectation: E(n;F),
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ─→ B[x],
natural_number: $n,
equal: s = t ∈ T,
rationals: ℚ
Lemmas :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
int_seg_wf,
random-variable_wf,
nat_wf,
finite-prob-space_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,
lt_int_wf,
bool_wf,
equal-wf-base,
assert_wf,
le_int_wf,
le_wf,
bnot_wf,
expectation-constant,
int-subtype-rationals,
top_wf,
subtype_rel_dep_function,
length_wf,
rationals_wf,
p-outcome_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
int_subtype_base,
expectation-rv-add,
Error :qsum_wf,
decidable__lt,
le-add-cancel2,
lelt_wf,
not-le-2,
subtract-is-less,
qadd_wf,
expectation_wf,
iff_weakening_equal,
squash_wf,
true_wf,
subtype_rel-int_seg,
add-mul-special,
zero-mul,
subtype_rel_self
\mforall{}[k:\mBbbN{}]. \mforall{}[p:FinProbSpace]. \mforall{}[n:\mBbbN{}]. \mforall{}[X:\mBbbN{}k {}\mrightarrow{} RandomVariable(p;n)].
(E(n;\mlambda{}s.\mSigma{}0 \mleq{} i < k. X i s) = \mSigma{}0 \mleq{} i < k. E(n;X i))
Date html generated:
2015_07_17-AM-07_59_00
Last ObjectModification:
2015_02_03-PM-09_44_38
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