hcju/proofwikips huggingface dataset:
Dedekind's Theorem/Corollary
Dedekind's Theorem/Corollary 0 83566 441930 2020-01-01T22:30:08Z Prime.mover 59 Created page with "== Theorem ==  Let $\tuple {L, R}$ be a Dedekind cut of the set of real numbers $\R$.   Then either $L$ con..." wikitext text/x-wiki == Theorem ==  Let $\tuple {L, R}$ be a Dedekind cut of the set of real numbers $\R$. Then either $L$ contains a largest number or $R$ contains a smallest number.  == Proof == From Dedekind's Theorem, there exists a unique real number such that: :$l \le \gamma$ for all $l \in L$ :$\gamma \le r$ for all $r \in R$. Let $\gamma \in L$. Then by definition $\gamma$ is the largest number in $L$. Let $\gamma \in R$. Then by definition $\gamma$ is the smallest number in $R$. By the definition of Dedekind cut, $\tuple {L, R}$ is a partition of $\R$. Hence $\gamma$ is either in $L$ or $R$, but not both. That is, $\gamma$ is either: :the largest number in $L$ or: :the smallest number in $R$. Hence the result.  == Sources == * {{BookReference|Principles of Mathematical Analysis|1964|ed = 2nd|edpage = Second Edition|Walter Rudin|prev = Dedekind's Theorem|next = Dedekind's Theorem/Proof 3}}: Chapter $1$: The Real and Complex Number Systems: Real Numbers: $1.32$. Corollary
Uniqueness of Positive Root of Positive Real Number/Negative Exponent
Uniqueness of Positive Root of Positive Real Number/Negative Exponent 0 83583 442142 2020-01-03T21:18:38Z Prime.mover 59 Created page with "== Theorem ==  Let $x \in \R$ be a real number such that $x > 0$.  Let $n \in \Z$ be an integer such that $n < 0..." wikitext text/x-wiki == Theorem ==  Let $x \in \R$ be a real number such that $x > 0$. Let $n \in \Z$ be an integer such that $n < 0$. Then there is at most one $y \in \R: y \ge 0$ such that $y^n = x$.  == Proof == Let $m = -n$. Let $g$ be the real function defined on $\hointr 0 \to$ defined by: :$\map g y = y^m$ From the definition of power: :$\map g y = \dfrac 1 {\map f y}$ Hence $\map g y$ is strictly decreasing. {{explain|Needs to invoke a result about reciprocals inverting the sense.}} From Strictly Monotone Mapping with Totally Ordered Domain is Injective: :there is at most one $y \in \R: y \ge 0$ such that $y^n = x$.
Russell's Paradox/Corollary
== Corollary to Russell's Paradox ==  :$\not \exists x: \forall y: \paren {\map \RR {x, y} \iff \neg \map \RR {y, y} }$  Given a relation $\RR$, there cannot exist an element $x$ that bears $\RR$ to all $y$ that do not bear $\RR$ to $y$. == Proof ==  there does exist such an $x$. Let $\RR$ be such that $\map \RR {x, x}$. Then $\neg \map \RR {x, x}$. Hence it cannot be the case that $\map \RR {x, x}$. Now suppose that $\neg \map \RR {x, x}$. Then by definition of $x$ it follows that $\map \RR {x, x}$. In both cases a contradiction results. Hence there can be no such $x$.  == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Barber Paradox|next = Axiom:Axiom of Specification/Set Theory}}: Chapter $1$: General Background: $\S 8$ Russell's paradox
Axiom:Axiom of Extension/Class Theory
Axiom:Axiom of Extension/Class Theory 100 83823 444630 444266 2020-01-23T20:48:15Z Prime.mover 59 Prime.mover moved page Axiom:Axiom of Extension/Classes to Axiom:Axiom of Extension/Class Theory wikitext text/x-wiki == Axiom ==  Let $A$ and $B$ be classes. Then: :$\forall x: \paren {x \in A \iff x \in B} \iff A = B$  Hence the order in which the elements are listed in the sets is immaterial. == Also known as == {{:Axiom:Axiom of Extension/Also known as}} == Also see == * Definition:Class Equality * Definition:Equals == Linguistic Note == {{:Axiom:Axiom of Extension/Linguistic Note}} == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Definition:Subclass|next = Definition:Class Equality/Definition 2}}: Chapter $2$: Some Basics of Class-Set Theory: $\S 1$ Extensionality and separation Extension Extension Extension d49c7lcjvs32dhy04hu4mqht7odovxl
Axiom:Axiom of Extension/Set Theory
== Axiom ==  Let $A$ and $B$ be sets. The '''axiom of extension''' states that $A$ and $B$ are equal  they contain the same elements. That is, : :every element of $A$ is also an element of $B$ and: :every element of $B$ is also an element of $A$. This can be formulated as follows:  === Formulation 1 === {{:Axiom:Axiom of Extension/Set Theory/Formulation 1}} === Formulation 2 === In set theories that define $=$ instead of admitting it as a primitive, the '''axiom of extension''' can be formulated as: {{:Axiom:Axiom of Extension/Set Theory/Formulation 2}} The order of the elements in the sets is immaterial. Hence a set is completely and uniquely determined by its elements. == Also known as == {{:Axiom:Axiom of Extension/Also known as}} == Also see == * Definition:Set Equality * Definition:Equals * Axiom:Axiom of Extension (Classes) == Linguistic Note == {{:Axiom:Axiom of Extension/Linguistic Note}} == Sources == * {{BookReference|Naive Set Theory|1960|Paul R. Halmos|prev = Definition:Set Equality/Definition 1|next = Definition:Subset}}: $\S 1$: The Axiom of Extension * {{BookReference|Abstract Algebra|1964|W.E. Deskins|prev = Definition:Uniqueness of Set Elements|next = Axiom:Axiom of Extension/Also known as}}: Chapter $1$: A Common Language: $\S 1.1$ Sets * {{BookReference|Elements of Abstract Algebra|1966|Richard A. Dean|prev = Definition:Set Equality/Definition 1|next = Definition:Explicit Set Definition}}: $\S 0.2$. Sets * {{BookReference|The Joy of Sets: Fundamentals of Contemporary Set Theory|1993|Keith Devlin|ed = 2nd|edpage = Second Edition|prev = Definition:Set Equality/Definition 1|next = Axiom:Axiom of Extension/Set Theory/Formulation 1}}: $\S 1$: Naive Set Theory: $\S 1.1$: What is a Set? Extension Extension Extension Extension ezzwxmd0yb8s2uo5azlvo4angx795ly
Axiom:Axiom of Specification/Class Theory
== Axiom == The '''axiom of specification''' is an axiom schema which can be formally stated as follows:  Let $\map \phi {A_1, A_2, \ldots, A_n, x}$ be a function of propositional logic such that: :$A_1, A_2, \ldots, A_n$ are a finite number of free variables whose domain ranges over all classes :$x$ is a free variable whose domain ranges over all sets. Then the '''axiom of specification''' gives that: :$\forall A_1, A_2, \ldots, A_n: \exists B: \forall x: \paren {x \in B \iff \paren {x \in B \land \phi {A_1, A_2, \ldots, A_n, x} } }$ where each of $B$ ranges over arbitrary classes.  This means that for any finite number $A_1, A_2, \ldots, A_n$ of subclasses of the universal class $V$, the class  $B$ exists (or can be formed) of all sets $x \in V$ that satisfy the function $\map \phi {A_1, A_2, \ldots, A_n, x}$. == Also known as == {{:Axiom:Axiom of Specification/Also known as}} == Also see == * Axiom:Axiom of Specification/Set Theory == Historical Note == {{:Axiom:Axiom of Specification/Historical Note}} {{Languages|Axiom of specification}} {{language|German|Aussonderungsaxiom|lit = axiom of segregation}} {{end-languages}} == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Axiom:Axiom of Specification/Also known as|next = Not Every Class is a Set/Proof 1}}: Chapter $2$: Some Basics of Class-Set Theory: $\S 1$ Extensionality and separation Specification Specification Specification 4ucgl8faz73dvwupsgvw7quikszyn4c
Axiom:Axiom of Transitivity
== Axiom == Let $V$ be a basic universe.  :$V$ is a transitive class.  That is, every set $S$ which is an element of $V$ is a subclass of $V$. Briefly: :Every set is a class. == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Definition:Basic Universe Axioms|next = Axiom:Axiom of Swelledness}}: Chapter $2$: Some Basics of Class-Set Theory: $\S 2$ Transitivity and supercompleteness Transitivity scdhzlpwz76qf6x9m0n8b032aanphlz
Axiom:Axiom of Swelledness
== Axiom == Let $V$ be a basic universe.  :$V$ is a swelled class.  That is, every subclass of a set which is an element of $V$ is a set in $V$. Briefly: :Every subclass of a set is a set. == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Axiom:Axiom of Transitivity|next = Basic Universe is Supercomplete}}: Chapter $2$: Some Basics of Class-Set Theory: $\S 2$ Transitivity and supercompleteness Swelledness qts7d0jpsbn35fi3ipb1mrhwouvkz41
Axiom:Axiom of Empty Set/Class Theory
== Axiom == Let $V$ be a basic universe.  The empty class $\O$ is a set, that is: :$\O \in V$  == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Empty Class is Supercomplete|next = Basic Universe is not Empty}}: Chapter $2$: Some Basics of Class-Set Theory: $\S 3$ Axiom of the empty set Empty Set Empty Set Empty Set 5q3ryuct4vv49oes7o7j8g6p3i4z23x
Axiom:Axiom of Pairing/Set Theory/Formulation 1
== Axiom ==  For any two sets, there exists a set to which only those two sets are elements: :$\forall A: \forall B: \exists x: \forall y: \paren {y \in x \iff y = A \lor y = B}$ Thus it is possible to create a set that contains as elements any two sets that have already been created.  == Also known as == {{:Axiom:Axiom of Pairing/Also known as}} == Also see == * Equivalence of Definitions of Axiom of Pairing == Sources == * {{BookReference|Point Set Topology|1964|Steven A. Gaal|prev = Set Definition by Predicate/Examples/Sums of Two Squares|next = Definition:Doubleton}}: Introduction to Set Theory: $1$. Elementary Operations on Sets * {{MathWorld|Zermelo-Fraenkel Axioms|Zermelo-FraenkelAxioms}} * {{MathWorld|Axiom of the Unordered Pair|AxiomoftheUnorderedPair}} Pairing 0vps8rk1gp0i13mvkkqa7sam5szgj7u
Axiom:Axiom of Pairing/Set Theory/Formulation 2
== Axiom ==  For any two sets, there exists a set containing those two sets as elements: :$\forall A: \forall B: \exists x: \forall y: \paren {y \in x \implies y = A \lor y = B}$ Thus it is possible to create a set that contains as elements '''at least''' two sets that have already been created.  == Also known as == {{:Axiom:Axiom of Pairing/Also known as}} == Also see == * Equivalence of Definitions of Axiom of Pairing == Sources == * {{BookReference|Naive Set Theory|1960|Paul R. Halmos|prev = Empty Set is Subset of All Sets/Proof 2|next = Definition:Doubleton}}: $\S 3$: Unordered Pairs Pairing 4kdr8tfqdpzc7yqjhyixitdy2i6nkbp
Axiom:Axiom of Pairing/Class Theory
== Axiom == === Formulation 1 === {{:Axiom:Axiom of Pairing/Class Theory/Formulation 1}} === Formulation 2 === {{:Axiom:Axiom of Pairing/Class Theory/Formulation 2}} == Also known as == {{:Axiom:Axiom of Pairing/Also known as}} == Also see == * Equivalence of Definitions of Axiom of Pairing for Classes * Definition:Doubleton Class * Doubleton Class can be Formed from Two Sets, which demonstrates the fact that the class $\set {a, b}$ can be created in the first place Pairing Pairing Pairing Pairing 0dvx4rpabm6uigw4fmmjdmfferroicn
Axiom:Axiom of Pairing/Class Theory/Formulation 1
== Axiom ==  Let $a$ and $b$ be sets. Then the class $\set {a, b}$ is likewise a set.  == Also known as == {{:Axiom:Axiom of Pairing/Also known as}} == Also see == * Equivalence of Definitions of Axiom of Pairing for Classes == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Doubleton Class of Equal Sets is Singleton Class|next = Axiom:Axiom of Pairing/Class Theory/Formulation 2}}: Chapter $2$: Some Basics of Class-Set Theory: $\S 4$ The pairing axiom Pairing b0jm1h7muszglmtaayg9uzvuujle2at
Axiom:Axiom of Pairing/Class Theory/Formulation 2
== Axiom ==  Let $a$ and $b$ be sets. Then there exists a set $c$ such that $a \in c$ and $b \in c$.  == Also known as == {{:Axiom:Axiom of Pairing/Also known as}} == Also see == * Equivalence of Definitions of Axiom of Pairing for Classes == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Axiom:Axiom of Pairing/Class Theory/Formulation 1|next = Equivalence of Definitions of Axiom of Pairing for Classes}}: Chapter $2$: Some Basics of Class-Set Theory: $\S 4$ The pairing axiom Pairing b0z2difchl5tiyxr25667hu45jlh9bi
Axiom:Axiom of Unions/Class Theory
== Axiom ==  Let $x$ be a set (of sets). Then its union $\displaystyle \bigcup x$ is also a set.  == Also known as == {{:Axiom:Axiom of Unions/Also known as}} == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Definition:Set Union/General Definition|next = Definition:Intersection of Class}}: Chapter $2$: Some Basics of Class-Set Theory: $\S 5$ The union axiom Unions Unions Unions Unions 1r5hxyqtplv3nii7ppvijkwqih0qqga
Axiom:Axiom of Powers/Class Theory
== Axiom ==  Let $x$ be a set. Then its power set $\powerset x$ is also a set.  == Also known as == {{:Axiom:Axiom of Powers/Also known as}} == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Power Set Exists and is Unique|next = Axiom:Axiom of Powers/Set Theory}}: Chapter $2$: Some Basics of Class-Set Theory: $\S 6$ The power axiom :::''although there exists a misprint: it is referred to as the '''Power aet axiom'''.'' Powers Powers Powers Powers lbb7ba64npqzd7u1kflb2nwjvfmvn82
Chain Rule
 * Chain Rule for Derivatives * Chain Rule for Probability ropx1clmxaww5s5fmzg03to31b4a8kc
Projection from Product Topology is Continuous/General Result/Proof
Projection from Product Topology is Continuous/General Result/Proof 0 84475 449480 2020-02-16T13:02:03Z Prime.mover 59 Created page with "== Theorem == {{:Projection from Product Topology is Continuous/General Result}}  == Proof ==  By definition of the Definition:Tychonoff Topology|Tychonoff topo..." wikitext text/x-wiki == Theorem == {{:Projection from Product Topology is Continuous/General Result}} == Proof ==  By definition of the Tychonoff topology on $S$: :$\tau$ is the initial topology on $S$ with respect to $\family {\pr_i}_{i \mathop \in I}$ By definition of the Initial Topoplogy:Definition 2: :$\tau$ is the coarsest topology on $S$ such that each $\pr_i: S \to S_i$ is a $\struct{\tau, \tau_i}$-continuous.
Projection from Product Topology is Open/General Result/Proof
Projection from Product Topology is Open/General Result/Proof 0 84476 449481 2020-02-16T13:08:46Z Prime.mover 59 Created page with "== Theorem == {{:Projection from Product Topology is Open/General Result}}  == Proof ==  Let $U \in \tau$.  It follows from the definition of Definition:Tychono..." wikitext text/x-wiki == Theorem == {{:Projection from Product Topology is Open/General Result}} == Proof ==  Let $U \in \tau$. It follows from the definition of Tychonoff topology that $U$ can be expressed as: :$\displaystyle U = \bigcup_{j \mathop \in J} \bigcap_{k \mathop = 1}^{n_j} \map {\pr_{i_{k, j} }^{-1} } {U_{k, j} }$ where: :$J$ is an arbitrary index set :$n_j \in \N$ :$i_{k, j} \in I$ :$U_{k, j} \in \tau_{i_{k, j} }$. For all $i' \in I$, define $V_{i', k, j} \in \tau_{i'}$ by: :$V_{i', k, j} = \begin {cases} U_{k, j} & : i' = i_{k, j} \\ S_{i'} & : i' \ne i_{k, j} \end {cases}$ For all $i \in I$, we have: {{begin-eqn}} {{eqn | l = \map {\pr_i} U       | r = \bigcup_{j \mathop \in J} \map {\pr_i} {\bigcap_{k \mathop = 1}^{n_j} \map {\pr_{i_{k,j} }^{-1} } { U_{k,j} } }       | c = Image of Union under Relation: Family of Sets }} {{eqn | r = \bigcup_{j \mathop \in J} \map {\pr_i} {\bigcap_{k \mathop = 1}^{n_j} \prod_{i' \mathop \in I} V_{i', k, j} }       | c = {{Defof|Projection (Mapping Theory)|Projection}} }} {{eqn | r = \bigcup_{j \mathop \in J} \map {\pr_i} {\prod_{i' \mathop \in I} \bigcap_{k \mathop = 1}^{n_j} V_{i', k, j} }       | c = Cartesian Product of Intersections: General Case }} {{eqn | r = \bigcup_{j \mathop \in J} \bigcap_{k \mathop = 1}^{n_j} V_{i,k,j}       | c = {{Defof|Projection (Mapping Theory)|Projection}} }} {{end-eqn}} As: :$\displaystyle \bigcup_{j \mathop \in J} \bigcap_{k \mathop = 1}^{n_j} V_{i, k, j} \in \tau_i$ it follows that $\pr_i$ is open.
Axiom:Axiom of Infinity/Class Theory
== Axiom == === Formulation 1 === {{:Axiom:Axiom of Infinity/Class Theory/Formulation 1}} === Formulation 2 === {{:Axiom:Axiom of Infinity/Class Theory/Formulation 2}} === Formulation 3 === {{:Axiom:Axiom of Infinity/Class Theory/Formulation 3}} == Also see == * Equivalence of Formulations of Axiom of Infinity for Zermelo Universe Infinity Infinity Infinity klx7xc3gld1khpe2k745q44yanxdsls
Axiom:Axiom of Infinity/Class Theory/Formulation 1
== Axiom ==  Let $\omega$ be the class of natural numbers as constructed by the Von Neumann construction: {{begin-eqn}} {{eqn | l = 0       | o = :=       | r = \O       | c =  }} {{eqn | l = 1       | o = :=       | r = 0 \cup \set 0       | c =  }} {{eqn | l = 2       | o = :=       | r = 1 \cup \set 1       | c =  }} {{eqn | l = 3       | o = :=       | r = 2 \cup \set 2       | c =  }} {{eqn | o = \vdots       | c =  }} {{eqn | l = n + 1       | o = :=       | r = n \cup \set n       | c =  }} {{eqn | o = \vdots       | c =  }} {{end-eqn}} Then $\omega$ is a set.  == Also see == * Equivalence of Formulations of Axiom of Infinity for Zermelo Universe == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Definition:Inductive Set Definition for Natural Numbers|next = Definition:Zermelo Universe}}: Chapter $3$: The Natural Numbers: $\S 2$ Definition of the Natural Numbers: Axiom $A_7$ Infinity Infinity mhsvoa9flwdf1bidqd73qx0ix0q0dsl
Axiom:Axiom of Infinity/Class Theory/Formulation 2
== Axiom ==  There exists an inductive set.  == Also see == * Equivalence of Formulations of Axiom of Infinity for Zermelo Universe == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Definition:Zermelo Universe|next = Axiom:Axiom of Infinity/Class Theory/Formulation 3/Historical Note}}: Chapter $3$: The Natural Numbers: $\S 2$ Definition of the Natural Numbers Infinity Infinity lu67jeskhv591puclw4kt2ka86u3h0s
Axiom:Axiom of Infinity/Class Theory/Formulation 3
== Axiom ==  Not every set is a natural number.  == Also see == * Equivalence of Formulations of Axiom of Infinity for Zermelo Universe == Historical Note == {{:Axiom:Axiom of Infinity/Class Theory/Formulation 3/Historical Note}} == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Axiom:Axiom of Infinity/Class Theory/Formulation 3/Historical Note|next = Equivalence of Formulations of Axiom of Infinity for Zermelo Universe}}: Chapter $3$: The Natural Numbers: $\S 2$ Definition of the Natural Numbers Infinity Infinity 5rohtd602vj2snp31dz1wzeorlmvcax
Condition for Expectation of Non-Negative Random Variable to be Zero
Condition for Expectation of Non-Negative Random Variable to be Zero 0 85335 455455 2020-03-18T11:29:40Z Caliburn 3218 Created page with "== Theorem ==  Let $X$ be a random variable.   Let:   :$\map \Pr {X \ge 0} = 1$   Then $\expect X = 0$  $\map \Pr {X = 0} =..." wikitext text/x-wiki == Theorem ==  Let $X$ be a random variable.  Let:  :$\map \Pr {X \ge 0} = 1$ Then $\expect X = 0$  $\map \Pr {X = 0} = 1$.  == Proof ==
Covariance of Linear Combination of Random Variables with Another
Covariance of Linear Combination of Random Variables with Another 0 85343 455483 2020-03-18T12:17:31Z Caliburn 3218 Created page with "== Theorem ==  Let $X, Y, Z$ be random variables.   Let $a, b$ be real numbers.   Then:   :$\cov {a X +..." wikitext text/x-wiki == Theorem ==  Let $X, Y, Z$ be random variables.  Let $a, b$ be real numbers. Then:  :$\cov {a X + b Y, Z} = a \cov {X, Z} + b \cov {Y, Z}$  == Proof == {{begin-eqn}} {{eqn	| l = \cov {a X + b Y, Z} 	| r = \expect {\paren {a X + b Y} Z} - \expect {a X + b Y} \expect Z 	| c = Covariance as Expectation of Product minus Product of Expectations }} {{eqn	| r = a \expect {X Z} + b \expect {Y Z} - \paren {a \expect X + b \expect Y} \expect Z 	| c = Linearity of Expectation Function }} {{eqn	| r = a \paren {\expect {X Z} - \expect X \expect Z} + b \paren {\expect {Y Z} - \expect Y \expect Z} }} {{eqn	| r = a \cov {X, Z} + b \cov {Y, Z} 	| c = Covariance as Expectation of Product minus Product of Expectations }} {{end-eqn}}
Covariance of Sums of Random Variables/Lemma
Covariance of Sums of Random Variables/Lemma 0 85345 455505 2020-03-18T14:14:08Z Caliburn 3218 Created page with "== Theorem ==  Let $n$ be a strictly positive integer.  Let $\sequence {X_i}_{1 \le i \le n}$ be a Definition:Sequence|..." wikitext text/x-wiki == Theorem ==  Let $n$ be a strictly positive integer. Let $\sequence {X_i}_{1 \le i \le n}$ be a sequence of random variables. Let $Y$ be a random variable. Then:  :$\displaystyle \cov {\sum_{i \mathop = 1}^n X_i, Y} = \sum_{i \mathop = 1}^n \cov {X_i, Y}$   == Proof == Proof by induction: For all $n \in \N$, let $\map P n$ be the proposition:  :$\displaystyle \cov {\sum_{i \mathop = 1}^n X_i, Y} = \sum_{i \mathop = 1}^n \cov {X_i, Y}$ === Basis for the Induction === We have that:  :$\displaystyle \cov {\sum_{i \mathop = 1}^n X_i, Y} = \cov {X_1, Y} = \sum_{i \mathop = 1}^1 \cov {X_i, Y}$ We therefore have that $\map P 1$ is true.  This is our base case. === Induction Hypothesis === Suppose that $\map P n$ is true for some fixed $n \in \N$.  That is:  :$\displaystyle \cov {\sum_{i \mathop = 1}^n X_i, Y} = \sum_{i \mathop = 1}^n \cov {X_i, Y}$ We aim to show that it logically follows that $\map P {n + 1}$ is true.  That is:  :$\displaystyle \cov {\sum_{i \mathop = 1}^{n + 1} X_i, Y} = \sum_{i \mathop = 1}^{n + 1} \cov {X_i, Y}$ === Induction Step === This is our induction step: We have:  {{begin-eqn}} {{eqn	| l = \cov {\sum_{i \mathop = 1}^{n + 1} X_i, Y} 	| r = \cov {\sum_{i \mathop = 1}^n X_i + X_{n + 1}, Y} 	| c = splitting up the sum }} {{eqn	| r = \cov {\sum_{i \mathop = 1}^n X_i, Y} + \cov {X_{n + 1}, Y} 	| c = Covariance of Linear Combination of Random Variables with Another }} {{eqn	| r = \sum_{i \mathop = 1}^n \cov {X_i, Y} + \cov {X_{n + 1}, Y} 	| c = induction hypothesis }} {{eqn	| r = \sum_{i \mathop = 1}^{n + 1} \cov {X_i, Y} }} {{end-eqn}} Hence the result by induction.
Expectation of Linear Transformation of Random Variable
Expectation of Linear Transformation of Random Variable 0 85348 455529 2020-03-18T16:10:35Z Caliburn 3218 Created page with "== Theorem ==  Let $X$ be a random variable.   Let $a, b$ be real numbers.    Then we have:   :$\expect..." wikitext text/x-wiki == Theorem ==  Let $X$ be a random variable.  Let $a, b$ be real numbers.  Then we have:  :$\expect {a X + b} = a \expect X + b$ where $\expect X$ denotes the expectation of $X$.   == Proof == === Discrete Random Variable === {{:Expectation of Linear Transformation of Random Variable/Discrete}} === Continuous Random Variable === {{:Expectation of Linear Transformation of Random Variable/Continuous}}