DIST_1: Probability on Finite and Discrete Set and Uniform Distribution

:: Probability on Finite and Discrete Set and Uniform Distribution
:: by Hiroyuki Okazaki
::
:: Received May 5, 2009
:: Copyright (c) 2009-2021 Association of Mizar Users

notation

let S be set ;
let s be FinSequence of S;
synonym whole_event s for dom S;

end;

notation

let S be set ;
let s be FinSequence of S;
let x be set ;
synonym event_pick (x,s) for Coim (S,s);

end;

definition

let S be set ;
let s be FinSequence of S;
let x be set ;
:: original: event_pick
redefine func event_pick (x,s) -> Event of (whole_event );
correctness by RELAT_1:132;

end;

definition

let S be finite set ;
let s be FinSequence of S;
let x be set ;

func frequency (x,s) -> Nat equals :: DIST_1:def 1
card (event_pick (x,s));

end;

:: deftheorem defines frequency DIST_1:def 1 :

theorem :: DIST_1:1

canceled;

::$CT

theorem :: DIST_1:2

for S being set
for s being FinSequence of S
for e being set st e in whole_event holds
ex x being Element of S st e in event_pick (x,s)

proof end;

definition

let S be finite set ;
let s be FinSequence of S;
let x be set ;

func FDprobability (x,s) -> Real equals :: DIST_1:def 2
(frequency (x,s)) / (len s);

end;

:: deftheorem defines FDprobability DIST_1:def 2 :

theorem :: DIST_1:3

canceled;

::$CT

theorem :: DIST_1:4

for S being finite set
for s being FinSequence of S
for x being set holds frequency (x,s) = (len s) * (FDprobability (x,s))

proof end;

definition

let S be finite set ;
let s be FinSequence of S;

func FDprobSEQ s -> FinSequence of REAL means :Def3: :: DIST_1:def 3
( dom it = Seg (card S) & ( for n being Nat st n in dom it holds
it . n = FDprobability (((canFS S) . n),s) ) );

proof end;

proof end;

end;

:: deftheorem Def3 defines FDprobSEQ DIST_1:def 3 :

theorem :: DIST_1:5

for S being finite set
for s being FinSequence of S
for x being set holds FDprobability (x,s) = prob (event_pick (x,s)) by CARD_1:62;

definition

let S be finite set ;
let s, t be FinSequence of S;

pred s,t -are_prob_equivalent means :: DIST_1:def 4
for x being set holds FDprobability (x,s) = FDprobability (x,t);

reflexivity ;
symmetry ;

end;

:: deftheorem defines -are_prob_equivalent DIST_1:def 4 :

theorem Th4: :: DIST_1:6

for S being finite set
for s, t, u being FinSequence of S st s,t -are_prob_equivalent & t,u -are_prob_equivalent holds
s,u -are_prob_equivalent

proof end;

definition

let S be finite set ;
let s be FinSequence of S;

func Finseq-EQclass s -> Subset of (S *) equals :: DIST_1:def 5
{ t where t is FinSequence of S : s,t -are_prob_equivalent } ;

proof end;

end;

:: deftheorem defines Finseq-EQclass DIST_1:def 5 :

registration

let S be finite set ;
let s be FinSequence of S;

cluster Finseq-EQclass s -> non empty ;

proof end;

end;

theorem Th5: :: DIST_1:7

for S being finite set
for s, t being FinSequence of S holds
( s,t -are_prob_equivalent iff t in Finseq-EQclass s )

proof end;

theorem Th6: :: DIST_1:8

for S being finite set
for s being FinSequence of S holds s in Finseq-EQclass s ;

theorem Th7: :: DIST_1:9

for S being finite set
for s, t being FinSequence of S holds
( s,t -are_prob_equivalent iff Finseq-EQclass s = Finseq-EQclass t )

proof end;

definition

let S be finite set ;
defpred S₁[ set ] means ex u being FinSequence of S st $1 = Finseq-EQclass u;

func distribution_family S -> Subset-Family of (S *) means :Def6: :: DIST_1:def 6
for A being Subset of (S *) holds
( A in it iff ex s being FinSequence of S st A = Finseq-EQclass s );

proof end;

proof end;

end;

:: deftheorem Def6 defines distribution_family DIST_1:def 6 :

registration

let S be finite set ;

cluster distribution_family S -> non empty ;

proof end;

end;

theorem Th8: :: DIST_1:10

for S being finite set
for s, t being FinSequence of S holds
( s,t -are_prob_equivalent iff FDprobSEQ s = FDprobSEQ t )

proof end;

theorem :: DIST_1:11

for S being finite set
for s, t being FinSequence of S st t in Finseq-EQclass s holds
FDprobSEQ s = FDprobSEQ t

proof end;

definition

let S be finite set ;

func GenProbSEQ S -> Function of (distribution_family S),(REAL *) means :Def7: :: DIST_1:def 7
for x being Element of distribution_family S ex s being FinSequence of S st
( s in x & it . x = FDprobSEQ s );

proof end;

proof end;

end;

:: deftheorem Def7 defines GenProbSEQ DIST_1:def 7 :

theorem Th10: :: DIST_1:12

for S being finite set
for s being FinSequence of S holds (GenProbSEQ S) . (Finseq-EQclass s) = FDprobSEQ s

proof end;

registration

let S be finite set ;

cluster GenProbSEQ S -> one-to-one ;

proof end;

end;

definition

let S be finite set ;
let p be ProbFinS FinSequence of REAL ;
assume A1: ex s being FinSequence of S st FDprobSEQ s = p ;

func distribution (p,S) -> Element of distribution_family S means :Def8: :: DIST_1:def 8
(GenProbSEQ S) . it = p;

proof end;

proof end;

end;

:: deftheorem Def8 defines distribution DIST_1:def 8 :

definition

let S be finite set ;
let s be FinSequence of S;

func freqSEQ s -> FinSequence of NAT means :Def9: :: DIST_1:def 9
( dom it = Seg (card S) & ( for n being Nat st n in dom it holds
it . n = (len s) * ((FDprobSEQ s) . n) ) );

proof end;

proof end;

end;

:: deftheorem Def9 defines freqSEQ DIST_1:def 9 :

theorem Th11: :: DIST_1:13

for S being non empty finite set
for s being non empty FinSequence of S
for n being Nat st n in Seg (card S) holds
ex x being Element of S st
( (freqSEQ s) . n = frequency (x,s) & x = (canFS S) . n )

proof end;

theorem Th12: :: DIST_1:14

for S being finite set
for s being FinSequence of S holds freqSEQ s = (len s) * (FDprobSEQ s)

proof end;

theorem Th13: :: DIST_1:15

for S being finite set
for s being FinSequence of S holds Sum (freqSEQ s) = (len s) * (Sum (FDprobSEQ s))

proof end;

theorem :: DIST_1:16

for S being non empty finite set
for s being non empty FinSequence of S
for n being Nat st n in dom s holds
ex m being Nat st
( (freqSEQ s) . m = frequency ((s . n),s) & s . n = (canFS S) . m )

proof end;

Lm1: for S being Function
for X being set st S is disjoint_valued holds
S | X is disjoint_valued

proof end;

Lm2: for n being Nat
for S being Function
for L being FinSequence of NAT st S is disjoint_valued & dom S = dom L & n = len L & ( for i being Nat st i in dom S holds
( S . i is finite & L . i = card (S . i) ) ) holds
( Union S is finite & card (Union S) = Sum L )

proof end;

theorem Th15: :: DIST_1:17

for S being Function
for L being FinSequence of NAT st S is disjoint_valued & dom S = dom L & ( for i being Nat st i in dom S holds
( S . i is finite & L . i = card (S . i) ) ) holds
( Union S is finite & card (Union S) = Sum L )

proof end;

theorem Th16: :: DIST_1:18

for S being non empty finite set
for s being non empty FinSequence of S holds Sum (freqSEQ s) = len s

proof end;

theorem Th17: :: DIST_1:19

for S being non empty finite set
for s being non empty FinSequence of S holds Sum (FDprobSEQ s) = 1

proof end;

registration

let S be non empty finite set ;
let s be non empty FinSequence of S;

cluster FDprobSEQ s -> ProbFinS ;

proof end;

end;

definition

let S be non empty finite set ;

mode distProbFinS of S -> ProbFinS FinSequence of REAL means :Def10: :: DIST_1:def 10
( len it = card S & ex s being FinSequence of S st FDprobSEQ s = it );

proof end;

end;

:: deftheorem Def10 defines distProbFinS DIST_1:def 10 :

theorem Th18: :: DIST_1:20

for S being non empty finite set
for p being distProbFinS of S holds
( distribution (p,S) is Element of distribution_family S & (GenProbSEQ S) . (distribution (p,S)) = p )

proof end;

definition

let S be finite set ;
let s be FinSequence of S;

attr s is uniformly_distributed means :: DIST_1:def 11
for n being Nat st n in dom (FDprobSEQ s) holds
(FDprobSEQ s) . n = 1 / (card S);

end;

:: deftheorem defines uniformly_distributed DIST_1:def 11 :

theorem Th19: :: DIST_1:21

for S being finite set
for s being FinSequence of S st s is uniformly_distributed holds
FDprobSEQ s is constant

proof end;

theorem Th20: :: DIST_1:22

for S being finite set
for s, t being FinSequence of S st s is uniformly_distributed & s,t -are_prob_equivalent holds
t is uniformly_distributed

proof end;

theorem Th21: :: DIST_1:23

for S being finite set
for s, t being FinSequence of S st s is uniformly_distributed & t is uniformly_distributed holds
s,t -are_prob_equivalent

proof end;

registration

let S be finite set ;

cluster canFS S -> uniformly_distributed for FinSequence of S;

proof end;

end;

Lm3: for S being finite set
for s being FinSequence of S st s in Finseq-EQclass (canFS S) holds
s is uniformly_distributed
by Th5, Th20;

Lm4: for S being finite set
for s being FinSequence of S st s is uniformly_distributed holds
for t being FinSequence of S st t is uniformly_distributed holds
t in Finseq-EQclass s

proof end;

definition

let S be finite set ;

func uniform_distribution S -> Element of distribution_family S means :Def12: :: DIST_1:def 12
for s being FinSequence of S holds
( s in it iff s is uniformly_distributed );

proof end;

proof end;

end;

:: deftheorem Def12 defines uniform_distribution DIST_1:def 12 :

registration

let S be non empty finite set ;

cluster Relation-like NAT -defined REAL -valued non empty Function-like constant finite V47() V48() V49() FinSequence-like FinSubsequence-like V77() ProbFinS for distProbFinS of S;

proof end;

end;

definition

let S be non empty finite set ;

func Uniform_FDprobSEQ S -> distProbFinS of S equals :: DIST_1:def 13
FDprobSEQ (canFS S);

proof end;

end;

:: deftheorem defines Uniform_FDprobSEQ DIST_1:def 13 :

registration

let S be non empty finite set ;

cluster Uniform_FDprobSEQ S -> constant ;

coherence by Th19;

end;

theorem :: DIST_1:24

for S being non empty finite set holds uniform_distribution S = distribution ((Uniform_FDprobSEQ S),S)

proof end;