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,
nat: ℕ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T,
qmul: r * s,
qadd: r + s,
rationals: ℚ
Lemmas :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
rationals_wf,
random-variable_wf,
qadd_wf,
qmul_wf,
null-seq_wf,
int_seg_wf,
length_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,
nat_wf,
finite-prob-space_wf,
eq_int_wf,
bool_wf,
equal-wf-base,
int_subtype_base,
assert_wf,
le_weakening,
bnot_wf,
not_wf,
weighted-sum_wf2,
squash_wf,
true_wf,
p-outcome_wf,
expectation_wf,
not-le-2,
rv-shift-linear,
rv-shift_wf,
iff_weakening_equal,
rv-add_wf,
rv-scale_wf,
ws-linear,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot
\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:
2015_07_17-AM-07_58_53
Last ObjectModification:
2015_02_03-PM-09_44_23
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