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Theorem List for Metamath Proof Explorer - 24701-24800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremprobtot 24701 The Probbiliy of the universe set is 1 (Second axiom of Kolmogorov) (Contributed by Thierry Arnoux, 8-Dec-2016.)
Prob

Theoremprob01 24702 A Probbiliy is bounded in [ 0 , 1 ] (First axiom of Kolmogorov) (Contributed by Thierry Arnoux, 25-Dec-2016.)
Prob

Theoremprobnul 24703 The Probbiliy of the empty event set is 0. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Prob

Theoremunveldomd 24704 The universe is an element of the domain of the probability, the universe (entire probability space) being in our construction. (Contributed by Thierry Arnoux, 22-Jan-2017.)
Prob

Theoremunveldom 24705 The universe is an element of the domain of the probability, the universe (entire probability space) being in our construction. (Contributed by Thierry Arnoux, 22-Jan-2017.)
Prob

Theoremnuleldmp 24706 The empty set is an element of the domain of the probability. (Contributed by Thierry Arnoux, 22-Jan-2017.)
Prob

Theoremprobcun 24707* The probability of the union of a countable disjoint set of events is the sum of their probabilities. (Third axiom of Kolmogorov) Here, the construct cannot be used as it can handle infinite indexing set only if they are subsets of , which is not the case here. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Prob Disj Σ*

Theoremprobun 24708 The probability of the union two incompatible events is the sum of their probabilities. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Prob

Theoremprobdif 24709 The probabiliy of the difference of two event sets (Contributed by Thierry Arnoux, 12-Dec-2016.)
Prob

Theoremprobinc 24710 A probabiliy law is increasing with regard to event set inclusion. (Contributed by Thierry Arnoux, 10-Feb-2017.)
Prob

Theoremprobdsb 24711 The probability of the complement of a set. That is, the probability that the event does not occur. (Contributed by Thierry Arnoux, 15-Dec-2016.)
Prob

Theoremprobmeasd 24712 A probability measure is a measure. (Contributed by Thierry Arnoux, 2-Feb-2017.)
Prob       measures

Theoremprobvalrnd 24713 The value of a probability is a real number. (Contributed by Thierry Arnoux, 2-Feb-2017.)
Prob

Theoremprobtotrnd 24714 The probability of the universe set is finite. (Contributed by Thierry Arnoux, 2-Feb-2017.)
Prob

Theoremtotprobd 24715* Law of total probability, deduction form. (Contributed by Thierry Arnoux, 25-Dec-2016.)
Prob                                   Disj        Σ*

Theoremtotprob 24716* Law of total probability (Contributed by Thierry Arnoux, 25-Dec-2016.)
Prob Disj Σ*

TheoremprobfinmeasbOLD 24717* Build a probability measure from a finite measure (Contributed by Thierry Arnoux, 17-Dec-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
measures /𝑒 Prob

Theoremprobfinmeasb 24718 Build a probability measure from a finite measure (Contributed by Thierry Arnoux, 31-Jan-2017.)
measures 𝑓/𝑐 /𝑒 Prob

Theoremprobmeasb 24719* Build a probability from a measure and a set with finite measure (Contributed by Thierry Arnoux, 25-Dec-2016.)
measures Prob

19.3.15.2  Conditional Probabilities

Syntaxccprob 24720 Extends class notation with the conditional probability builder.
cprob

Definitiondf-cndprob 24721* Define the conditional probability. (Contributed by Thierry Arnoux, 14-Sep-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
cprob Prob

Theoremcndprobval 24722 The value of the conditional probability , i.e. the probability for the event , given , under the probability law . (Contributed by Thierry Arnoux, 21-Jan-2017.)
Prob cprob

Theoremcndprobin 24723 An identity linking conditional probability and intersection. (Contributed by Thierry Arnoux, 13-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
Prob cprob

Theoremcndprob01 24724 The conditional probability has values in . (Contributed by Thierry Arnoux, 13-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
Prob cprob

Theoremcndprobtot 24725 The conditional probability given a certain event is one. (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
Prob cprob

Theoremcndprobnul 24726 The conditional probability given empty event is zero. (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
Prob cprob

Theoremcndprobprob 24727* The conditional probability defines a probability law. (Contributed by Thierry Arnoux, 23-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
Prob cprob Prob

Theorembayesth 24728 Bayes Theorem (Contributed by Thierry Arnoux, 20-Dec-2016.) (Revised by Thierry Arnoux, 21-Jan-2017.)
Prob cprob cprob

19.3.15.3  Real Valued Random Variables

Syntaxcrrv 24729 Extend class notation with the class of real valued random variables.
rRndVar

Definitiondf-rrv 24730 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 Prob MblFnM𝔅

Theoremrrvmbfm 24731 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.)
Prob       rRndVar MblFnM𝔅

Theoremisrrvv 24732* Elementhood to the set of real-valued random variables with respect to the probability . (Contributed by Thierry Arnoux, 25-Jan-2017.)
Prob       rRndVar 𝔅

Theoremrrvvf 24733 A real-valued random variable is a function. (Contributed by Thierry Arnoux, 25-Jan-2017.)
Prob       rRndVar

Theoremrrvfn 24734 A real-valued random variable is a function over the universe. (Contributed by Thierry Arnoux, 25-Jan-2017.)
Prob       rRndVar

Theoremrrvdm 24735 The domain of a random variable is the universe. (Contributed by Thierry Arnoux, 25-Jan-2017.)
Prob       rRndVar

Theoremrrvrnss 24736 The range of a random variable as a subset of . (Contributed by Thierry Arnoux, 6-Feb-2017.)
Prob       rRndVar

Theoremrrvf2 24737 A real-valued random variable is a function. (Contributed by Thierry Arnoux, 25-Jan-2017.)
Prob       rRndVar

Theoremrrvdmss 24738 The domain of a random variable. This is useful to shorten proofs. (Contributed by Thierry Arnoux, 25-Jan-2017.)
Prob       rRndVar

Theoremrrvfinvima 24739* For a real-value random variable , any open interval in is the image of a measurable set. (Contributed by Thierry Arnoux, 25-Jan-2017.)
Prob       rRndVar       𝔅

Theorem0rrv 24740* The constant function equal to zero is a random variable. (Contributed by Thierry Arnoux, 16-Jan-2017.) (Revised by Thierry Arnoux, 30-Jan-2017.)
Prob       rRndVar

Theoremrrvadd 24741 The sum of two random variables is a random variable (Contributed by Thierry Arnoux, 4-Jun-2017.)
Prob       rRndVar       rRndVar       rRndVar

Theoremrrvmulc 24742 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.)
Prob       rRndVar              𝑓/𝑐 rRndVar

Theoremrrvsum 24743 An indexed sum of random variables is a random variable. (Contributed by Thierry Arnoux, 22-May-2017.)
Prob       rRndVar              rRndVar

19.3.15.4  Preimage set mapping operator

Syntaxcorvc 24744 Extend class notation to include the preimage set mapping operator.
RV/𝑐

Definitiondf-orvc 24745* Define the preimage set mapping operator. In probability theory, the notation denotes the probability that a random variable takes the value . We introduce here an operator which enables to write this in Metamath as RV/𝑐 , and keep a similar notation. Because with this notation RV/𝑐 is a set, we can also apply it to conditional probabilities, like in RV/𝑐 RV/𝑐 .

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

The most commonly used relations are: - equality: as RV/𝑐 cf. ideq 5054- elementhood: as RV/𝑐 cf. epel 4526- less-than: as RV/𝑐

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

Theoremorvcval 24746* Value of the preimage mapping operator applied on a given random variable and constant (Contributed by Thierry Arnoux, 19-Jan-2017.)
RV/𝑐

Theoremorvcval2 24747* Another way to express the value of the preimage mapping operator (Contributed by Thierry Arnoux, 19-Jan-2017.)
RV/𝑐

Theoremelorvc 24748* Elementhood of a preimage (Contributed by Thierry Arnoux, 21-Jan-2017.)
RV/𝑐

Theoremorvcval4 24749* The value of the preimage mapping operator can be restricted to preimages in the base set of the topology. Cf. orvcval 24746 (Contributed by Thierry Arnoux, 21-Jan-2017.)
sigAlgebra              MblFnMsigaGen              RV/𝑐

Theoremorvcoel 24750* If the relation produces open sets, preimage maps by a measurable function are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
sigAlgebra              MblFnMsigaGen                     RV/𝑐

Theoremorvccel 24751* If the relation produces closed sets, preimage maps by a measurable function are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
sigAlgebra              MblFnMsigaGen                     RV/𝑐

Theoremelorrvc 24752* Elementhood of a preimage for a real-valued random variable. (Contributed by Thierry Arnoux, 21-Jan-2017.)
Prob       rRndVar              RV/𝑐

Theoremorrvcval4 24753* The value of the preimage mapping operator can be restricted to preimages of subsets of RR. (Contributed by Thierry Arnoux, 21-Jan-2017.)
Prob       rRndVar              RV/𝑐

Theoremorrvcoel 24754* If the relation produces open sets, preimage maps of a random variable are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
Prob       rRndVar                     RV/𝑐

Theoremorrvccel 24755* If the relation produces closed sets, preimage maps are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
Prob       rRndVar                     RV/𝑐

Theoremorvcgteel 24756 Preimage maps produced by the "greater than or equal" relation are measurable sets. (Contributed by Thierry Arnoux, 5-Feb-2017.)
Prob       rRndVar              RV/𝑐

19.3.15.5  Distribution Functions

Theoremorvcelval 24757 Preimage maps produced by the "elementhood" relation (Contributed by Thierry Arnoux, 6-Feb-2017.)
Prob       rRndVar       𝔅       RV/𝑐

Theoremorvcelel 24758 Preimage maps produced by the "elementhood" relation are measurable sets. (Contributed by Thierry Arnoux, 5-Feb-2017.)
Prob       rRndVar       𝔅       RV/𝑐

Theoremdstrvval 24759* The value of the distribution of a random variable. (Contributed by Thierry Arnoux, 9-Feb-2017.)
Prob       rRndVar       𝔅 RV/𝑐        𝔅

Theoremdstrvprob 24760* The distribution of a random variable is a probability law (TODO: could be shortened using dstrvval 24759) (Contributed by Thierry Arnoux, 10-Feb-2017.)
Prob       rRndVar       𝔅 RV/𝑐        Prob

19.3.15.6  Cumulative Distribution Functions

Theoremorvclteel 24761 Preimage maps produced by the "lower than or equal" relation are measurable sets. (Contributed by Thierry Arnoux, 4-Feb-2017.)
Prob       rRndVar              RV/𝑐

Theoremdstfrvel 24762 Elementhood of preimage maps produced by the "lower than or equal" relation. (Contributed by Thierry Arnoux, 13-Feb-2017.)
Prob       rRndVar                            RV/𝑐

Theoremdstfrvunirn 24763* The limit of all preimage maps by the "lower than or equal" relation is the universe. (Contributed by Thierry Arnoux, 12-Feb-2017.)
Prob       rRndVar       RV/𝑐

Theoremorvclteinc 24764 Preimage maps produced by the "lower than or equal" relation are increasing. (Contributed by Thierry Arnoux, 11-Feb-2017.)
Prob       rRndVar                            RV/𝑐 RV/𝑐

Theoremdstfrvinc 24765* A cumulative distribution function is non-decreasing. (Contributed by Thierry Arnoux, 11-Feb-2017.)
Prob       rRndVar       RV/𝑐

Theoremdstfrvclim1 24766* The limit of the cumulative distribution function is one. (Contributed by Thierry Arnoux, 12-Feb-2017.) (Revised by Thierry Arnoux, 11-Jul-2017.)
Prob       rRndVar       RV/𝑐

19.3.15.7  Probabilities - example

Theoremcoinfliplem 24767 Division in the extended real numbers can be used for the coin-flip example. (Contributed by Thierry Arnoux, 15-Jan-2017.)
𝑓/𝑐               𝑓/𝑐 /𝑒

Theoremcoinflipprob 24768 The we defined for coin-flip is a probability law. (Contributed by Thierry Arnoux, 15-Jan-2017.)
𝑓/𝑐               Prob

Theoremcoinflipspace 24769 The space of our coin-flip probability (Contributed by Thierry Arnoux, 15-Jan-2017.)
𝑓/𝑐

Theoremcoinflipuniv 24770 The universe of our coin-flip probability is . (Contributed by Thierry Arnoux, 15-Jan-2017.)
𝑓/𝑐

Theoremcoinfliprv 24771 The we defined for coin-flip is a random variable. (Contributed by Thierry Arnoux, 12-Jan-2017.)
𝑓/𝑐               rRndVar

Theoremcoinflippv 24772 The probability of heads is one-half. (Contributed by Thierry Arnoux, 15-Jan-2017.)
𝑓/𝑐

Theoremcoinflippvt 24773 The probability of tails is one-half. (Contributed by Thierry Arnoux, 5-Feb-2017.)
𝑓/𝑐

19.3.15.8  Bertrand's Ballot Problem

Theoremballotlemoex 24774* is a set. (Contributed by Thierry Arnoux, 7-Dec-2016.)

Theoremballotlem1 24775* The size of the universe is a binomial coefficient. (Contributed by Thierry Arnoux, 23-Nov-2016.)

Theoremballotlemelo 24776* Elementhood in . (Contributed by Thierry Arnoux, 17-Apr-2017.)

Theoremballotlem2 24777* The probability that the first vote picked in a count is a B (Contributed by Thierry Arnoux, 23-Nov-2016.)

Theoremballotlemfval 24778* The value of F. (Contributed by Thierry Arnoux, 23-Nov-2016.)

Theoremballotlemfelz 24779* has values in . (Contributed by Thierry Arnoux, 23-Nov-2016.)

Theoremballotlemfp1 24780* If the th ballot is for A, goes up 1. If the th ballot is for B, goes down 1. (Contributed by Thierry Arnoux, 24-Nov-2016.)

Theoremballotlemfc0 24781* takes value 0 between negative and positive values. (Contributed by Thierry Arnoux, 24-Nov-2016.)

Theoremballotlemfcc 24782* takes value 0 between positive and negative values. (Contributed by Thierry Arnoux, 2-Apr-2017.)

Theoremballotlemfmpn 24783* finishes counting at . (Contributed by Thierry Arnoux, 25-Nov-2016.)

Theoremballotlemfval0 24784* always starts counting at 0 . (Contributed by Thierry Arnoux, 25-Nov-2016.)

Theoremballotleme 24785* Elements of . (Contributed by Thierry Arnoux, 14-Dec-2016.)

Theoremballotlemodife 24786* Elements of . (Contributed by Thierry Arnoux, 7-Dec-2016.)

Theoremballotlem4 24787* If the first pick is a vote for B, A is not ahead throughout the count (Contributed by Thierry Arnoux, 25-Nov-2016.)

Theoremballotlem5 24788* If A is not ahead throughout, there is a where votes are tied. (Contributed by Thierry Arnoux, 1-Dec-2016.)

Theoremballotlemi 24789* Value of for a given counting . (Contributed by Thierry Arnoux, 1-Dec-2016.)

Theoremballotlemiex 24790* Properties of . (Contributed by Thierry Arnoux, 12-Dec-2016.)

Theoremballotlemi1 24791* The first tie cannot be reached at the first pick. (Contributed by Thierry Arnoux, 12-Mar-2017.)

Theoremballotlemii 24792* The first tie cannot be reached at the first pick. (Contributed by Thierry Arnoux, 4-Apr-2017.)

Theoremballotlemsup 24793* The set of zeroes of satisfies the conditions to have a supremum (Contributed by Thierry Arnoux, 1-Dec-2016.)

Theoremballotlemimin 24794* is the first tie. (Contributed by Thierry Arnoux, 1-Dec-2016.)

Theoremballotlemic 24795* If the first vote is for B, the vote on the first tie is for A. (Contributed by Thierry Arnoux, 1-Dec-2016.)

Theoremballotlem1c 24796* If the first vote is for A, the vote on the first tie is for B. (Contributed by Thierry Arnoux, 4-Apr-2017.)

Theoremballotlemsval 24797* Value of (Contributed by Thierry Arnoux, 12-Apr-2017.)

Theoremballotlemsv 24798* Value of evaluated at for a given counting . (Contributed by Thierry Arnoux, 12-Apr-2017.)

Theoremballotlemsgt1 24799* maps values less than to values greater than 1. (Contributed by Thierry Arnoux, 28-Apr-2017.)

Theoremballotlemsdom 24800* Domain of for a given counting . (Contributed by Thierry Arnoux, 12-Apr-2017.)

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