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Theorem List for Metamath Proof Explorer - 23601-23700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-ibfm 23601* Define the set of integrable functions with respect to a measure  m. (Contributed by Thierry Arnoux, 16-Jan-2017.)
 |- IblFnM  =  ( m  e.  U. ran measures  |->  { f  e.  (MblFnM `  <. dom  m , 𝔅 >.
 )  |  (𝜇 U. dom  m  f  _d m  e.  RR  /\  ran  f  C_  ( 0 [,)  +oo ) ) }
 )
 
TheoremitgmvolTMP 23602* When the Lebesgue integral is used, the measure integral is the Lebesgue integral. This is currently a placeholder and a complete (and complex) proof will be necessary once the measure integral will be properly defined. (Contributed by Thierry Arnoux, 26-Jan-2017.)
 |-𝜇 A F  _d vol  =  S. A ( F `
  x )  _d x
 
TheoremitgmcntTMP 23603* When the counting measure is used, the measure integral is the usual sum operation. This is currently a placeholder and a complete proof will be necessary once the measure integral will be properly defined. (Contributed by Thierry Arnoux, 27-Jan-2017.)
 |-  ( O  e.  V  ->𝜇 A F  _d ( #  |`  ~P O )  = Σ* x  e.  A ( F `  x ) )
 
Theoremitgmeq123dTMP 23604 Equality deduction for measure integral. (Contributed by Thierry Arnoux, 27-Jan-2017.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  M  =  P )   =>    |-  ( ph  ->𝜇 A F  _d M  =𝜇 B G  _d P )
 
Theoremitgmeq1dTMP 23605 Equality deduction for measure integral. (Contributed by Thierry Arnoux, 27-Jan-2017.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->𝜇 A F  _d M  =𝜇 B F  _d M )
 
Theoremitgmeq2dTMP 23606 Equality deduction for measure integral. (Contributed by Thierry Arnoux, 27-Jan-2017.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ph  ->𝜇 A F  _d M  =𝜇 A G  _d M )
 
Theoremitgmeq3dTMP 23607 Equality deduction for measure integral. (Contributed by Thierry Arnoux, 27-Jan-2017.)
 |-  ( ph  ->  M  =  P )   =>    |-  ( ph  ->𝜇 A F  _d M  =𝜇 A F  _d P )
 
Theoremisibfm 23608 The property of being integrable with respect to a given measure  M. (Contributed by Thierry Arnoux, 27-Mar-2017.)
 |-  ( M  e.  U. ran measures  ->  ( F  e.  (IblFnM `  M ) 
 <->  ( F  e.  (MblFnM ` 
 <. dom  M , 𝔅
 >. )  /\  ran  F  C_  ( 0 [,)  +oo )  /\𝜇 U. dom  M  F  _d M  e.  RR )
 ) )
 
18.3.40  Indicator Functions
 
Syntaxcind 23609 Extend class notation with the indicator function generator.
 class 𝟭
 
Definitiondf-ind 23610* Define the indicator function generator. (Contributed by Thierry Arnoux, 20-Jan-2017.)
 |- 𝟭  =  ( o  e.  _V  |->  ( a  e.  ~P o  |->  ( x  e.  o  |->  if ( x  e.  a ,  1 ,  0 ) ) ) )
 
Theoremindv 23611* Value of the indicator function generator with domain  O. (Contributed by Thierry Arnoux, 23-Aug-2017.)
 |-  ( O  e.  V  ->  (𝟭 `  O )  =  ( a  e.  ~P O  |->  ( x  e.  O  |->  if ( x  e.  a ,  1 ,  0 ) ) ) )
 
Theoremindval 23612* Value of the indicator function generator for a set  A and a domain  O. (Contributed by Thierry Arnoux, 2-Feb-2017.)
 |-  (
 ( O  e.  V  /\  A  C_  O )  ->  ( (𝟭 `  O ) `  A )  =  ( x  e.  O  |->  if ( x  e.  A ,  1 ,  0 ) ) )
 
Theoremindval2 23613 Alternate value of the indicator function generator. (Contributed by Thierry Arnoux, 2-Feb-2017.)
 |-  (
 ( O  e.  V  /\  A  C_  O )  ->  ( (𝟭 `  O ) `  A )  =  ( ( A  X.  { 1 } )  u.  ( ( O  \  A )  X.  { 0 } ) ) )
 
Theoremindf 23614 An indicator function as a function with domain and codomain. (Contributed by Thierry Arnoux, 13-Aug-2017.)
 |-  (
 ( O  e.  V  /\  A  C_  O )  ->  ( (𝟭 `  O ) `  A ) : O --> { 0 ,  1 } )
 
Theoremindfval 23615 Value of the indicator function. (Contributed by Thierry Arnoux, 13-Aug-2017.)
 |-  (
 ( O  e.  V  /\  A  C_  O  /\  X  e.  O )  ->  ( ( (𝟭 `  O ) `  A ) `  X )  =  if ( X  e.  A ,  1 ,  0 ) )
 
Theorempr01ssre 23616 The range of the indicator function is a subset of  RR. (Contributed by Thierry Arnoux, 14-Aug-2017.)
 |-  { 0 ,  1 }  C_  RR
 
Theoremind1 23617 Value of the indicator function where it is  1. (Contributed by Thierry Arnoux, 14-Aug-2017.)
 |-  (
 ( O  e.  V  /\  A  C_  O  /\  X  e.  A )  ->  ( ( (𝟭 `  O ) `  A ) `  X )  =  1
 )
 
Theoremind0 23618 Value of the indicator function where it is  0. (Contributed by Thierry Arnoux, 14-Aug-2017.)
 |-  (
 ( O  e.  V  /\  A  C_  O  /\  X  e.  ( O  \  A ) )  ->  ( ( (𝟭 `  O ) `  A ) `  X )  =  0
 )
 
Theoremind1a 23619 Value of the indicator function where it is  1. (Contributed by Thierry Arnoux, 22-Aug-2017.)
 |-  (
 ( O  e.  V  /\  A  C_  O  /\  X  e.  O )  ->  ( ( ( (𝟭 `  O ) `  A ) `  X )  =  1  <->  X  e.  A ) )
 
Theoremindpi1 23620 Preimage of the singleton  { 1 } by the indicator function. See i1f1lem 19060. (Contributed by Thierry Arnoux, 21-Aug-2017.)
 |-  (
 ( O  e.  V  /\  A  C_  O )  ->  ( `' ( (𝟭 `  O ) `  A ) " { 1 } )  =  A )
 
Theoremindsum 23621* Finite sum of a product with the indicator function / cross-product with the indicator function. (Contributed by Thierry Arnoux, 14-Aug-2017.)
 |-  ( ph  ->  O  e.  Fin )   &    |-  ( ph  ->  A  C_  O )   &    |-  ( ( ph  /\  x  e.  O ) 
 ->  B  e.  CC )   =>    |-  ( ph  ->  sum_ x  e.  O  ( ( ( (𝟭 `  O ) `  A ) `  x )  x.  B )  =  sum_ x  e.  A  B )
 
Theoremindf1o 23622 The bijection between a power set and the set of indicator functions. (Contributed by Thierry Arnoux, 14-Aug-2017.)
 |-  ( O  e.  V  ->  (𝟭 `  O ) : ~P O
 -1-1-onto-> ( { 0 ,  1 }  ^m  O ) )
 
Theoremindpreima 23623 A function with range  { 0 ,  1 } as an indicator of the preimage of  { 1 } (Contributed by Thierry Arnoux, 23-Aug-2017.)
 |-  (
 ( O  e.  V  /\  F : O --> { 0 ,  1 } )  ->  F  =  ( (𝟭 `  O ) `  ( `' F " { 1 } ) ) )
 
Theoremindf1ofs 23624* The bijection between finite subsets and the indicator functions with finite support. (Contributed by Thierry Arnoux, 22-Aug-2017.)
 |-  ( O  e.  V  ->  ( (𝟭 `  O )  |`  Fin ) : ( ~P O  i^i  Fin ) -1-1-onto-> {
 f  e.  ( {
 0 ,  1 } 
 ^m  O )  |  ( `' f " { 1 } )  e.  Fin } )
 
18.3.41  Probability Theory
 
Syntaxcprb 23625 Extend class notation to include the class of probability measures.
 class Prob
 
Definitiondf-prob 23626 Define the class of probability measures as the set of measures with total measure 1. (Contributed by Thierry Arnoux, 14-Sep-2016.)
 |- Prob  =  { p  e.  U. ran measures  |  ( p `  U. dom  p )  =  1 }
 
Theoremelprob 23627 The property of being a probability measure (Contributed by Thierry Arnoux, 8-Dec-2016.)
 |-  ( P  e. Prob  <->  ( P  e.  U.
 ran measures  /\  ( P `  U.
 dom  P )  =  1 ) )
 
Theoremdomprobmeas 23628 A probability measure is a measure on its domain. (Contributed by Thierry Arnoux, 23-Dec-2016.)
 |-  ( P  e. Prob  ->  P  e.  (measures `  dom  P ) )
 
Theoremdomprobsiga 23629 The domain of a probability measure is a sigma-algebra. (Contributed by Thierry Arnoux, 23-Dec-2016.)
 |-  ( P  e. Prob  ->  dom  P  e.  U. ran sigAlgebra )
 
Theoremprobtot 23630 The Probbiliy of the universe set is 1 (Second axiom of Kolmogorov) (Contributed by Thierry Arnoux, 8-Dec-2016.)
 |-  ( P  e. Prob  ->  ( P `
  U. dom  P )  =  1 )
 
Theoremprob01 23631 A Probbiliy is bounded in [ 0 , 1 ] (First axiom of Kolmogorov) (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P )  ->  ( P `  A )  e.  ( 0 [,] 1 ) )
 
Theoremprobnul 23632 The Probbiliy of the empty event set is 0. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  ( P  e. Prob  ->  ( P `
  (/) )  =  0 )
 
Theoremunveldomd 23633 The universe is an element of the domain of the probability, the universe (entire probability space) being  U.
dom  P in our construction. (Contributed by Thierry Arnoux, 22-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   =>    |-  ( ph  ->  U. dom  P  e.  dom 
 P )
 
Theoremunveldom 23634 The universe is an element of the domain of the probability, the universe (entire probability space) being  U.
dom  P in our construction. (Contributed by Thierry Arnoux, 22-Jan-2017.)
 |-  ( P  e. Prob  ->  U. dom  P  e.  dom  P )
 
Theoremnuleldmp 23635 The empty set is an element of the domain of the probability. (Contributed by Thierry Arnoux, 22-Jan-2017.)
 |-  ( P  e. Prob  ->  (/)  e.  dom  P )
 
Theoremprobcun 23636* The probability of the union of a countable disjoint set of events is the sum of their probabilities. (Third axiom of Kolmogorov) Here, the  sum_ construct cannot be used as it can handle infinite indexing set only if they are subsets of 
ZZ, which is not the case here. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  (
 ( P  e. Prob  /\  A  e.  ~P dom  P  /\  ( A  ~<_  om  /\ Disj  x  e.  A x ) ) 
 ->  ( P `  U. A )  = Σ* x  e.  A ( P `  x ) )
 
Theoremprobun 23637 The probability of the union two incompatible events is the sum of their probabilities. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  ->  ( ( A  i^i  B )  =  (/)  ->  ( P `  ( A  u.  B ) )  =  ( ( P `  A )  +  ( P `  B ) ) ) )
 
Theoremprobdif 23638 The probabiliy of the difference of two event sets (Contributed by Thierry Arnoux, 12-Dec-2016.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  ->  ( P `  ( A 
 \  B ) )  =  ( ( P `
  A )  -  ( P `  ( A  i^i  B ) ) ) )
 
Theoremprobinc 23639 A probabiliy law is increasing with regard to event set inclusion. (Contributed by Thierry Arnoux, 10-Feb-2017.)
 |-  (
 ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  A  C_  B )  ->  ( P `  A )  <_  ( P `
  B ) )
 
Theoremprobdsb 23640 The probability of the complement of a set. That is, the probability that the event  A does not occur. (Contributed by Thierry Arnoux, 15-Dec-2016.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P )  ->  ( P `  ( U. dom  P  \  A ) )  =  ( 1  -  ( P `  A ) ) )
 
Theoremprobmeasd 23641 A probability measure is a measure. (Contributed by Thierry Arnoux, 2-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   =>    |-  ( ph  ->  P  e.  U. ran measures )
 
Theoremprobvalrnd 23642 The value of a probability is a real number. (Contributed by Thierry Arnoux, 2-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  A  e.  dom  P )   =>    |-  ( ph  ->  ( P `  A )  e. 
 RR )
 
Theoremprobtotrnd 23643 The probability of the universe set is finite. (Contributed by Thierry Arnoux, 2-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   =>    |-  ( ph  ->  ( P `  U.
 dom  P )  e.  RR )
 
Theoremtotprobd 23644* Law of total probability, deduction form. (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  A  e.  dom  P )   &    |-  ( ph  ->  B  e.  ~P dom  P )   &    |-  ( ph  ->  U. B  =  U. dom  P )   &    |-  ( ph  ->  B  ~<_  om )   &    |-  ( ph  -> Disj  b  e.  B b )   =>    |-  ( ph  ->  ( P `  A )  = Σ* b  e.  B ( P `
  ( b  i^i 
 A ) ) )
 
Theoremtotprob 23645* Law of total probability (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P  /\  ( U. B  =  U. dom  P  /\  B  e.  ~P
 dom  P  /\  ( B  ~<_ 
 om  /\ Disj  b  e.  B b ) ) ) 
 ->  ( P `  A )  = Σ* b  e.  B ( P `  ( b  i^i  A ) ) )
 
TheoremprobfinmeasbOLD 23646* Build a probability measure from a finite measure (Contributed by Thierry Arnoux, 17-Dec-2016.)
 |-  (
 ( M  e.  (measures `  S )  /\  ( M `  U. S )  e.  RR+ )  ->  ( x  e.  S  |->  ( ( M `  x ) /𝑒  ( M `  U. S ) ) )  e. Prob
 )
 
Theoremprobfinmeasb 23647 Build a probability measure from a finite measure (Contributed by Thierry Arnoux, 31-Jan-2017.)
 |-  (
 ( M  e.  (measures `  S )  /\  ( M `  U. S )  e.  RR+ )  ->  ( M𝑓/𝑐 /𝑒  ( M ` 
 U. S ) )  e. Prob )
 
Theoremprobmeasb 23648* Build a probability from a measure and a set with finite measure (Contributed by Thierry Arnoux, 25-Dec-2016.)
 |-  (
 ( M  e.  (measures `  S )  /\  A  e.  S  /\  ( M `
  A )  e.  RR+ )  ->  ( x  e.  S  |->  ( ( M `  ( x  i^i  A ) ) 
 /  ( M `  A ) ) )  e. Prob )
 
18.3.42  Conditional Probabilities
 
Syntaxccprob 23649 Extends class notation with the conditional probability builder.
 class cprob
 
Definitiondf-cndprob 23650* Define the conditional probability. (Contributed by Thierry Arnoux, 14-Sep-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
 |- cprob  =  ( p  e. Prob  |->  ( a  e.  dom  p ,  b  e.  dom  p  |->  ( ( p `  (
 a  i^i  b )
 )  /  ( p `  b ) ) ) )
 
Theoremcndprobval 23651 The value of the conditional probability , i.e. the probability for the event  A, given  B, under the probability law  P. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  ->  ( (cprob `  P ) `  <. A ,  B >. )  =  ( ( P `  ( A  i^i  B ) ) 
 /  ( P `  B ) ) )
 
Theoremcndprobin 23652 An identity linking conditional probability and intersection. (Contributed by Thierry Arnoux, 13-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
 |-  (
 ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  ( P `  B )  =/=  0
 )  ->  ( (
 (cprob `  P ) `  <. A ,  B >. )  x.  ( P `
  B ) )  =  ( P `  ( A  i^i  B ) ) )
 
Theoremcndprob01 23653 The conditional probability has values in  [ 0 ,  1 ]. (Contributed by Thierry Arnoux, 13-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
 |-  (
 ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  ( P `  B )  =/=  0
 )  ->  ( (cprob `  P ) `  <. A ,  B >. )  e.  (
 0 [,] 1 ) )
 
Theoremcndprobtot 23654 The conditional probability given a certain event is one. (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P  /\  ( P `  A )  =/=  0 )  ->  (
 (cprob `  P ) `  <. U. dom  P ,  A >. )  =  1 )
 
Theoremcndprobnul 23655 The conditional probability given empty event is zero. (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
 |-  (
 ( P  e. Prob  /\  A  e.  dom  P  /\  ( P `  A )  =/=  0 )  ->  (
 (cprob `  P ) `  <. (/) ,  A >. )  =  0 )
 
Theoremcndprobprob 23656* The conditional probability defines a probability law. (Contributed by Thierry Arnoux, 23-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
 |-  (
 ( P  e. Prob  /\  B  e.  dom  P  /\  ( P `  B )  =/=  0 )  ->  (
 a  e.  dom  P  |->  ( (cprob `  P ) `  <. a ,  B >. ) )  e. Prob )
 
Theorembayesth 23657 Bayes Theorem (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
 |-  (
 ( ( P  e. Prob  /\  A  e.  dom  P  /\  B  e.  dom  P )  /\  ( P `  A )  =/=  0  /\  ( P `  B )  =/=  0 )  ->  ( (cprob `  P ) `  <. A ,  B >. )  =  ( ( ( (cprob `  P ) `  <. B ,  A >. )  x.  ( P `
  A ) ) 
 /  ( P `  B ) ) )
 
18.3.43  Real Valued Random Variables
 
Syntaxcrrv 23658 Extend class notation with the class of real valued random variables.
 class rRndVar
 
Definitiondf-rrv 23659 In its generic definition, a random variable is a measurable function from a probability space to a Borel set. Here, we specifically target real-valued random variables, i.e. measurable function from a probability space to the Borel sigma algebra on the set of real numbers. (Contributed by Thierry Arnoux, 20-Sep-2016.) (Revised by Thierry Arnoux, 25-Jan-2017.)
 |- rRndVar  =  ( p  e. Prob  |->  (MblFnM `  <. dom  p , 𝔅 >.
 ) )
 
Theoremrrvmbfm 23660 A real-valued random variable is a measurable function from its sample space to the Borel Sigma Algebra. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   =>    |-  ( ph  ->  ( X  e.  (rRndVar `  P )  <->  X  e.  (MblFnM ` 
 <. dom  P , 𝔅
 >. ) ) )
 
Theoremisrrvv 23661* Elementhood to the set of real-valued random variables with respect to the probability  P. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   =>    |-  ( ph  ->  ( X  e.  (rRndVar `  P )  <->  ( X : U. dom  P --> RR  /\  A. y  e. 𝔅  ( `' X "
 y )  e.  dom  P ) ) )
 
Theoremrrvvf 23662 A real-valued random variable is a function. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  X : U. dom  P --> RR )
 
Theoremrrvfn 23663 A real-valued random variable is a function over the universe. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  X  Fn  U. dom  P )
 
Theoremrrvdm 23664 The domain of a random variable is the universe. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  dom 
 X  =  U. dom  P )
 
Theoremrrvrnss 23665 The range of a random variable as a subset of  RR. (Contributed by Thierry Arnoux, 6-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  ran 
 X  C_  RR )
 
Theoremrrvf2 23666 A real-valued random variable is a function. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  X : dom  X --> RR )
 
Theoremrrvdmss 23667 The domain of a random variable. This is useful to shorten proofs. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  U.
 dom  P  C_  dom  X )
 
Theoremrrvfinvima 23668* For a real-value random variable  X, any open interval in 
RR is the image of a measurable set. (Contributed by Thierry Arnoux, 25-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  A. y  e. 𝔅  ( `' X "
 y )  e.  dom  P )
 
Theorem0rrv 23669* The constant function equal to zero is a random variable. (Contributed by Thierry Arnoux, 16-Jan-2017.) (Revised by Thierry Arnoux, 30-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   =>    |-  ( ph  ->  ( x  e. 
 U. dom  P  |->  0 )  e.  (rRndVar `  P ) )
 
Theoremrrvmulc 23670 A random variable multiplied by a constant is a random variable. (Contributed by Thierry Arnoux, 17-Jan-2017.) (Revised by Thierry Arnoux, 22-May-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  C  e.  RR )   =>    |-  ( ph  ->  ( X𝑓/𝑐  x.  C )  e.  (rRndVar `  P ) )
 
18.3.44  Preimage set mapping operator
 
Syntaxcorvc 23671 Extend class notation to include the preimage set mapping operator.
 classRV/𝑐 R
 
Definitiondf-orvc 23672* Define the preimage set mapping operator. In probability theory, the notation  P ( X  =  A ) denotes the probability that a random variable  X takes the value  A. We introduce here an operator which enables to write this in Metamath as  ( P `  ( XRV/𝑐  _I  A ) ), and keep a similar notation. Because with this notation  ( XRV/𝑐  _I  A ) is a set, we can also apply it to conditional probabilities, like in  ( P `  ( XRV/𝑐  _I  A )  |  ( YRV/𝑐  _I  B ) ) ).

The oRVC operator transforms a relation  R into an operation taking a random variable  X and a constant  C, and returning the preimage through  X of the equivalence class of  C.

The most commonly used relations are: - equality:  { X  =  A } as  ( XRV/𝑐  _I  A ) cf. ideq 4852- elementhood:  { X  e.  A } as  ( XRV/𝑐  _E  A ) cf. epel 4324- less-than:  { X  <_  A } as  ( XRV/𝑐  <_  A )

Even though it is primarily designed to be used within probability theory and with random variables, this operator is defined on generic functions, and could be used in other fields, e.g. for continuous functions. (Contributed by Thierry Arnoux, 15-Jan-2017.)

 |-RV/𝑐 R  =  ( x  e.  { x  |  Fun  x } ,  a  e.  _V  |->  ( `' x " { y  |  y R a } )
 )
 
Theoremorvcval 23673* Value of the preimage mapping operator applied on a given random variable and constant (Contributed by Thierry Arnoux, 19-Jan-2017.)
 |-  ( ph  ->  Fun  X )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  A  e.  W )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  =  ( `' X " { y  |  y R A }
 ) )
 
Theoremorvcval2 23674* Another way to express the value of the preimage mapping operator (Contributed by Thierry Arnoux, 19-Jan-2017.)
 |-  ( ph  ->  Fun  X )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  A  e.  W )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  =  {
 z  e.  dom  X  |  ( X `  z
 ) R A }
 )
 
Theoremelorvc 23675* Elementhood of a preimage (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  Fun  X )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  A  e.  W )   =>    |-  ( ( ph  /\  z  e.  dom  X )  ->  ( z  e.  ( XRV/𝑐 R A )  <->  ( X `  z ) R A ) )
 
Theoremorvcval4 23676* The value of the preimage mapping operator can be restricted to preimages in the base set of the topology. Cf. orvcval 23673 (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  J  e.  Top )   &    |-  ( ph  ->  X  e.  (MblFnM ` 
 <. S ,  (sigaGen `  J ) >. ) )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  =  ( `' X " { y  e.  U. J  |  y R A } )
 )
 
Theoremorvcoel 23677* If the relation produces open sets, preimage maps by a measurable function are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  J  e.  Top )   &    |-  ( ph  ->  X  e.  (MblFnM ` 
 <. S ,  (sigaGen `  J ) >. ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  { y  e.  U. J  |  y R A }  e.  J )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  e.  S )
 
Theoremorvccel 23678* If the relation produces closed sets, preimage maps by a measurable function are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  S  e.  U. ran sigAlgebra )   &    |-  ( ph  ->  J  e.  Top )   &    |-  ( ph  ->  X  e.  (MblFnM ` 
 <. S ,  (sigaGen `  J ) >. ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  { y  e.  U. J  |  y R A }  e.  ( Clsd `  J )
 )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  e.  S )
 
Theoremelorrvc 23679* Elementhood of a preimage for a real-valued random variable. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ( ph  /\  z  e.  U. dom  P ) 
 ->  ( z  e.  ( XRV/𝑐 R A )  <->  ( X `  z ) R A ) )
 
Theoremorrvcval4 23680* The value of the preimage mapping operator can be restricted to preimages of subsets of RR. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  =  ( `' X " { y  e.  RR  |  y R A } ) )
 
Theoremorrvcoel 23681* If the relation produces open sets, preimage maps of a random variable are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  { y  e.  RR  |  y R A }  e.  ( topGen `
  ran  (,) ) )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  e.  dom  P )
 
Theoremorrvccel 23682* If the relation produces closed sets, preimage maps are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  { y  e.  RR  |  y R A }  e.  ( Clsd `  ( topGen `  ran  (,) ) ) )   =>    |-  ( ph  ->  ( XRV/𝑐 R A )  e.  dom  P )
 
Theoremorvcgteel 23683 Preimage maps produced by the "greater than or equal" relation are measurable sets. (Contributed by Thierry Arnoux, 5-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( XRV/𝑐 `' 
 <_  A )  e.  dom  P )
 
18.3.45  Distribution Functions
 
Theoremorvcelval 23684 Preimage maps produced by the "elementhood" relation (Contributed by Thierry Arnoux, 6-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e. 𝔅 )   =>    |-  ( ph  ->  ( XRV/𝑐  _E  A )  =  ( `' X " A ) )
 
Theoremorvcelel 23685 Preimage maps produced by the "elementhood" relation are measurable sets. (Contributed by Thierry Arnoux, 5-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e. 𝔅 )   =>    |-  ( ph  ->  ( XRV/𝑐  _E  A )  e.  dom  P )
 
Theoremdstrvval 23686* The value of the distribution of a random variable. (Contributed by Thierry Arnoux, 9-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  D  =  ( a  e. 𝔅 
 |->  ( P `  ( XRV/𝑐  _E  a ) ) ) )   &    |-  ( ph  ->  A  e. 𝔅 )   =>    |-  ( ph  ->  ( D `  A )  =  ( P `  ( `' X " A ) ) )
 
Theoremdstrvprob 23687* The distribution of a random variable is a probability law (TODO: could be shortened using dstrvval 23686) (Contributed by Thierry Arnoux, 10-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  D  =  ( a  e. 𝔅 
 |->  ( P `  ( XRV/𝑐  _E  a ) ) ) )   =>    |-  ( ph  ->  D  e. Prob )
 
18.3.46  Cumulative Distribution Functions
 
Theoremorvclteel 23688 Preimage maps produced by the "lower than or equal" relation are measurable sets. (Contributed by Thierry Arnoux, 4-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  ( XRV/𝑐  <_  A )  e.  dom  P )
 
Theoremdstfrvel 23689 Elementhood of preimage maps produced by the "lower than or equal" relation. (Contributed by Thierry Arnoux, 13-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  U. dom  P )   &    |-  ( ph  ->  ( X `  B )  <_  A )   =>    |-  ( ph  ->  B  e.  ( XRV/𝑐  <_  A ) )
 
Theoremdstfrvunirn 23690* The limit of all preimage maps by the "lower than or equal" relation is the universe. (Contributed by Thierry Arnoux, 12-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   =>    |-  ( ph  ->  U.
 ran  ( n  e. 
 NN  |->  ( XRV/𝑐  <_  n ) )  = 
 U. dom  P )
 
Theoremorvclteinc 23691 Preimage maps produced by the "lower than or equal" relation are increasing. (Contributed by Thierry Arnoux, 11-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A 
 <_  B )   =>    |-  ( ph  ->  ( XRV/𝑐  <_  A )  C_  ( XRV/𝑐  <_  B ) )
 
Theoremdstfrvinc 23692* A cumulative distribution function is non-decreasing. (Contributed by Thierry Arnoux, 11-Feb-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  F  =  ( x  e.  RR  |->  ( P `  ( XRV/𝑐  <_  x ) ) ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A  <_  B )   =>    |-  ( ph  ->  ( F `  A )  <_  ( F `  B ) )
 
Theoremdstfrvclim1 23693* The limit of the cumulative distribution function is one. (Contributed by Thierry Arnoux, 12-Feb-2017.) (Revised by Thierry Arnoux, 11-Jul-2017.)
 |-  ( ph  ->  P  e. Prob )   &    |-  ( ph  ->  X  e.  (rRndVar `  P ) )   &    |-  ( ph  ->  F  =  ( x  e.  RR  |->  ( P `  ( XRV/𝑐  <_  x ) ) ) )   =>    |-  ( ph  ->  F  ~~>  1 )
 
18.3.47  Probabilities - example
 
Theoremcoinfliplem 23694 Division in the extended real numbers can be used for the coin-flip example. (Contributed by Thierry Arnoux, 15-Jan-2017.)
 |-  H  e.  _V   &    |-  T  e.  _V   &    |-  H  =/=  T   &    |-  P  =  ( ( #  |`  ~P { H ,  T }
 )𝑓/𝑐  / 
 2 )   &    |-  X  =  { <. H ,  1 >. ,  <. T ,  0
 >. }   =>    |-  P  =  ( ( #  |`  ~P { H ,  T } )𝑓/𝑐 /𝑒  2 )
 
Theoremcoinflipprob 23695 The  P we defined for coin-flip is a probability law. (Contributed by Thierry Arnoux, 15-Jan-2017.)
 |-  H  e.  _V   &    |-  T  e.  _V   &    |-  H  =/=  T   &    |-  P  =  ( ( #  |`  ~P { H ,  T }
 )𝑓/𝑐  / 
 2 )   &    |-  X  =  { <. H ,  1 >. ,  <. T ,  0
 >. }   =>    |-  P  e. Prob
 
Theoremcoinflipspace 23696 The space of our coin-flip probability (Contributed by Thierry Arnoux, 15-Jan-2017.)
 |-  H  e.  _V   &    |-  T  e.  _V   &    |-  H  =/=  T   &    |-  P  =  ( ( #  |`  ~P { H ,  T }
 )𝑓/𝑐  / 
 2 )   &    |-  X  =  { <. H ,  1 >. ,  <. T ,  0
 >. }   =>    |- 
 dom  P  =  ~P { H ,  T }
 
Theoremcoinflipuniv 23697 The universe of our coin-flip probability is  { H ,  T }. (Contributed by Thierry Arnoux, 15-Jan-2017.)
 |-  H  e.  _V   &    |-  T  e.  _V   &    |-  H  =/=  T   &    |-  P  =  ( ( #  |`  ~P { H ,  T }
 )𝑓/𝑐  / 
 2 )   &    |-  X  =  { <. H ,  1 >. ,  <. T ,  0
 >. }   =>    |- 
 U. dom  P  =  { H ,  T }
 
Theoremcoinfliprv 23698 The  X we defined for coin-flip is a random variable. (Contributed by Thierry Arnoux, 12-Jan-2017.)
 |-  H  e.  _V   &    |-  T  e.  _V   &    |-  H  =/=  T   &    |-  P  =  ( ( #  |`  ~P { H ,  T }
 )𝑓/𝑐  / 
 2 )   &    |-  X  =  { <. H ,  1 >. ,  <. T ,  0
 >. }   =>    |-  X  e.  (rRndVar `  P )
 
Theoremcoinflippv 23699 The probability of heads is one-half. (Contributed by Thierry Arnoux, 15-Jan-2017.)
 |-  H  e.  _V   &    |-  T  e.  _V   &    |-  H  =/=  T   &    |-  P  =  ( ( #  |`  ~P { H ,  T }
 )𝑓/𝑐  / 
 2 )   &    |-  X  =  { <. H ,  1 >. ,  <. T ,  0
 >. }   =>    |-  ( P `  { H } )  =  (
 1  /  2 )
 
Theoremcoinflippvt 23700 The probability of tails is one-half. (Contributed by Thierry Arnoux, 5-Feb-2017.)
 |-  H  e.  _V   &    |-  T  e.  _V   &    |-  H  =/=  T   &    |-  P  =  ( ( #  |`  ~P { H ,  T }
 )𝑓/𝑐  / 
 2 )   &    |-  X  =  { <. H ,  1 >. ,  <. T ,  0
 >. }   =>    |-  ( P `  { T } )  =  (
 1  /  2 )
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