hcju/proofwikips huggingface dataset:
Definition:Field Homomorphism
Let $\struct {F, +, \times}$ and $\struct {K, \oplus, \otimes}$ be fields. Let $\phi: F \to K$ be a mapping such that both $+$ and $\times$ have the morphism property under $\phi$. That is, $\forall a, b \in F$: {{begin-eqn}} {{eqn | n = 1       | l = \map \phi {a + b}       | r = \map \phi a \oplus \map \phi b }} {{eqn | n = 2       | l = \map \phi {a \times b}       | r = \map \phi a \otimes \map \phi b }} {{end-eqn}} Then $\phi: \struct {F, +, \times} \to \struct {K, \oplus, \otimes}$ is a field homomorphism.
Definition:Monic Polynomial
Let $R$ be a commutative ring with unity. Let $f \in R \sqbrk x$ be a polynomial in one variable over $R$. Then $f$ is '''monic'''  $f$ is nonzero and its leading coefficient is $1$.
Definition:Leading Coefficient of Polynomial
Let $\struct {R, +, \circ}$ be a ring. Let $\struct {S, +, \circ}$ be a subring of $R$. Let $\displaystyle f = \sum_{k \mathop = 0}^n a_k \circ x^k$ be a polynomial in $x$ over $S$. The coefficient $a_n \ne 0_R$ is called the '''leading coefficient''' of $f$. === Polynomial Form === {{:Definition:Leading Coefficient of Polynomial/Polynomial Form}}
Definition:Experiment
An '''experiment''' can be defined in natural language as a '''test to see what happens'''. === Informal Definition === {{:Definition:Experiment/Informal Definition}} === Formal Definition === {{:Definition:Experiment/Formal Definition}}
Definition:Sample Space
Let $\EE$ be an experiment. The '''sample space''' of $\EE$ is usually denoted $\Omega$ (Greek capital '''omega'''), and is defined as '''the set of all possible outcomes of $\EE$'''.
Definition:Elementary Event
Let $\EE$ be an experiment. An '''elementary event''' of $\EE$, often denoted $\omega$ (Greek lowercase '''omega''') is one of the elements of the sample space $\Omega$ (Greek capital '''omega''') of $\EE$.
Definition:Event Space
Let $\EE$ be an experiment whose probability space is $\struct {\Omega, \Sigma, \Pr}$. The '''event space''' of $\EE$ is usually denoted $\Sigma$ (Greek capital '''sigma'''), and is '''the set of all outcomes of $\EE$ which are interesting'''. By definition, $\struct {\Omega, \Sigma}$ is a measurable space. Hence the '''event space''' $\Sigma$ is a sigma-algebra on $\Omega$.
Definition:Event
Let $\EE$ be an experiment. An '''event in $\EE$''' is an element of the event space $\Sigma$ of $\EE$.
Definition:Probability Measure
Let $\EE$ be an experiment. === Definition 1 === {{:Definition:Probability Measure/Definition 1}} === Definition 2 === {{:Definition:Probability Measure/Definition 2}} === Definition 3 === {{:Definition:Probability Measure/Definition 3}} === Definition 4 === {{:Definition:Probability Measure/Definition 4}}
Definition:Percentage
A '''percentage''' is a convenient way of expressing (usually) a fraction as an easily-grasped number, by multiplying it by $100$ and appending the symbol $\%$ which means '''per cent'''. As conventionally understood, the number being rendered as a '''percentage''' is between $0$ and $1$, thus determining the range of a '''percentage''' to be $0 \%$ to $100 \%$. However in many contexts that range is frequently extended (in particular, in the field of rhetoric): :''I want every member of this team to give one hundred and ten percent.''
Definition:Sigma-Ring
{{:Definition:Sigma-Ring/Definition 1}}
Definition:Delta-Ring
A '''delta-ring''' (which can conveniently be written '''$\delta$-ring''') is a ring of sets which is closed under countable intersections. That is, a ring of sets $\RR$ is a '''delta-ring''' : :$\displaystyle A_1, A_2, \ldots \in \RR \implies \bigcap_{n \mathop = 1}^\infty A_n \in \RR$
Definition:Delta-Algebra
A '''delta-algebra''' is a delta-ring with a unit. Thus, a '''delta-algebra''' is an algebra of sets which is closed under countable intersections.
Definition:Borel Algebra
A '''Borel algebra''' is sometimes used as another name for a $\sigma$-algebra or $\delta$-algebra. Can be confused with Borel $\sigma$-Algebra, which is a specific kind of $\sigma$-algebra.
Definition:Involution (Mapping)
An '''involution''' is a mapping which is its own inverse. === Definition 1 === {{:Definition:Involution (Mapping)/Definition 1}} === Definition 2 === {{:Definition:Involution (Mapping)/Definition 2}} === Definition 3 === {{:Definition:Involution (Mapping)/Definition 3}}
Definition:Additive Function (Conventional)
Let $f: S \to S$ be a mapping on an algebraic structure $\struct {S, +}$. Then $f$ is an '''additive function'''  it preserves the addition operation: :$\forall x, y \in S: \map f {x + y} = \map f x + \map f y$
Definition:Completely Additive Function
Let $\left({R, +, \times}\right)$ be a ring. Let $f: R \to R$ be a mapping on $R$. Then $f$ is described as '''completely additive''' : :$\forall m, n \in R: f \left({m \times n}\right) = f \left({m}\right) + f \left({n}\right)$ That is, a '''completely additive function''' is one where the value of a product of two numbers equals the sum of the value of each one individually.
Definition:Additive Function (Measure Theory)
Let $\SS$ be an algebra of sets. Let $f: \SS \to \overline \R$ be a function, where $\overline \R$ denotes the set of extended real numbers. Then $f$ is defined to be '''additive''' : :$\forall S, T \in \SS: S \cap T = \O \implies \map f {S \cup T} = \map f S + \map f T$ That is, for any two disjoint elements of $\SS$, $f$ of their union equals the sum of $f$ of the individual elements. Note from Finite Union of Sets in Additive Function that: :$\displaystyle \map f {\bigcup_{i \mathop = 1}^n S_i} = \sum_{i \mathop = 1}^n \map f {S_i}$ where $S_1, S_2, \ldots, S_n$ is any finite collection of pairwise disjoint elements of $\SS$.
Definition:Countably Additive Function
Let $\Sigma$ be a $\sigma$-algebra. Let $f: \Sigma \to \overline \R$ be a function, where $\overline \R$ denotes the set of extended real numbers. Then $f$ is defined as '''countably additive''' : :$\displaystyle \map f {\bigcup_{n \mathop \in \N} E_n} = \sum_{n \mathop \in \N} \map f {E_n}$ where $\left \langle {E_n} \right \rangle$ is any sequence of pairwise disjoint elements of $\Sigma$. That is, for any countably infinite set of pairwise disjoint elements of $\Sigma$, $f$ of their union equals the sum of $f$ of the individual elements.
Definition:Subadditive Function (Conventional)
Let $\left({S, +_S}\right)$ and $\left({T, +_T, \preceq}\right)$ be semigroups such that $\left({T, +_T, \preceq}\right)$ is ordered. Let $f: S \to T$ be a mapping from $S$ to $T$ which satisfies the relation: :$\forall a, b \in S: f \left({a +_S b}\right) \preceq f \left({a}\right) +_T f \left({b}\right)$ Then $f$ is defined as being '''subadditive'''. The usual context in which this is encountered is where $S$ and $T$ are both the set of real numbers $\R$ (or a subset of them).
Definition:Subadditive Function (Measure Theory)
Let $\mathcal S$ be an algebra of sets. Let $f: \mathcal S \to \overline \R$ be a function, where $\overline \R$ denotes the extended set of real numbers. Then $f$ is defined to be '''subadditive''' (or '''sub-additive''') iff: :$\forall S, T \in \mathcal S: f \left({S \cup T}\right) \le f \left({S}\right) + f \left({T}\right)$ That is, for any two elements of $\mathcal S$, $f$ applied to their union is not greater than the sum of $f$ of the individual elements.
Definition:Countably Subadditive Function
Let $\Sigma$ be a $\sigma$-algebra over a set $X$. Let $f: \Sigma \to \overline \R$ be a function, where $\overline \R$ denotes the set of extended real numbers. Then $f$ is defined as '''countably subadditive''' iff, for any sequence $\left \langle {E_n} \right \rangle_{n \in \N}$ of elements of $\Sigma$: :$\displaystyle f \left({\bigcup_{n \mathop = 0}^\infty E_n}\right) \le \sum_{n \mathop = 0}^\infty f \left({E_n}\right)$
Definition:Measurable Space
Let $\Sigma$ be a $\sigma$-algebra on a set $X$. Then the pair $\struct {X, \Sigma}$ is called a '''measurable space'''.
Definition:Monotone (Measure Theory)
Let $\mathcal S$ be an algebra of sets. Let $f: \mathcal S \to \overline \R$ be an extended real-valued function, where $\overline \R$ denotes the set of extended real numbers. Then $f$ is defined as '''monotone''' or '''monotonic''' iff: :$\forall A, B \in \mathcal S: A \subseteq B \iff f \left({A}\right) \le f \left({B}\right)$
Definition:Disjoint Union (Set Theory)
Let $\family {S_i}_{i \mathop \in I}$ be an $I$-indexed family of sets. The '''disjoint union''' of $\family {S_i}_{i \mathop \in I}$ is defined as the set: :$\displaystyle \bigsqcup_{i \mathop \in I} S_i = \bigcup_{i \mathop \in I} \set {\tuple {x, i}: x \in S_i}$ where $\bigcup$ denotes union. Each of the sets $S_i$ is canonically embedded in the '''disjoint union''' as the set: :$S_i^* = \set {\tuple {x, i}: x \in S_i}$
Definition:Disjoint Union (Probability Theory)
Let $\mathcal C$ be a collection of pairwise disjoint sets. That is, for all sets $A, B \in \mathcal C: A \ne B \implies A \cap B = \varnothing$. Then the union of all sets in $\mathcal C$ is called their '''disjoint union''': :$\displaystyle \bigsqcup_{A \mathop \in \mathcal C} A \equiv \bigcup_{A \mathop \in \mathcal C} A$ That is, in this context the term '''disjoint union''' means '''union of sets which are pairwise disjoint'''.
Definition:Equiprobable Outcomes
Let $\struct {\Omega, \Sigma, \Pr}$ be a finite probability space. Let $\Omega = \set {\omega_1, \omega_1, \ldots, \omega_n}$. Suppose that $\map \Pr {\omega_i} = \map \Pr {\omega_j}$ for all the $\omega_i, \omega_j \in \Omega$. Then from Probability Measure on Equiprobable Outcomes: :$\forall \omega \in \Omega: \map \Pr \omega = \dfrac 1 n$ :$\forall A \subseteq \Omega: \map \Pr A = \dfrac {\card A} n$ Such a probability space is said to have '''equiprobable outcomes''', and is sometimes referred to as an equiprobability space.
Definition:Logical Matrix
A '''logical matrix''' (or '''boolean matrix''') is a matrix whose entries are all either $0$ or $1$.
Definition:Finite Probability Space
A '''finite probability space''' is a discrete probability space $\left({\Omega, \Sigma, \Pr}\right)$ such that the sample space $\Omega$ is finite.  3fpbykva2knibxeo21kkyel1xvzyh8i
Definition:Die
thumb300pxrightTwo six-sided dice A '''die''' (plural: '''dice''') is a device for generating random numbers, traditionally between $1$ and $6$. A die usually consists of a cube on which each face denotes a different number, usually (again) from $1$ to $6$. Traditionally, opposite faces add to $7$, so: :$1$ is opposite $6$ :$2$ is opposite $5$ :$3$ is opposite $4$. The idea is that when you roll it, or throw it, or otherwise cause it to land upon a flattish surface, one of the sides will be face-up, and (with a good design of die) which side that is can not be precisely predicted. Hence it serves as a generator of a discrete uniform distribution. By using dice with different shapes (usually either platonic solids or prisms), a different probability space can be generated. {{DefinitionWanted|Describe general polyhedral dice, or link to a Wikipedia article on FRP or something.}}
Definition:Equiprobability Space
An '''equiprobability space''' is a finite probability space $\struct {\Omega, \Sigma, \Pr}$ with equiprobable outcomes. That is, for all $\omega_i, \omega_j \in \Omega$: :$\map \Pr {\omega_i} = \map \Pr {\omega_j}$ From Probability Measure on Equiprobable Outcomes, we have that: :$\forall \omega \in \Omega: \map \Pr \omega = \dfrac 1 n$ :$\forall A \subseteq \Omega: \map \Pr A = \dfrac {\card A} n$
Definition:Binary Notation
'''Binary notation''' is the technique of expressing numbers in base $2$. That is, every number $x \in \R$ is expressed in the form: :$\displaystyle x = \sum_{j \mathop \in \Z} r_j 2^j$ where $\forall j \in \Z: r_j \in \set {0, 1}$.
Definition:Hexadecimal Notation
'''Hexadecimal notation''' is the technique of expressing numbers in base $16$. Every number $x \in \R$ is expressed in the form: :$\displaystyle x = \sum_{j \mathop \in \Z} r_j 16^j$ where: :$\forall j \in \Z: r_j \in \left\{ {0, 1, \ldots, 15}\right\}$
Definition:General Logarithm
=== Positive Real Numbers === {{:Definition:General Logarithm/Positive Real}} === Complex Numbers === {{:Definition:General Logarithm/Complex}}
Definition:Conditional Probability
Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$. Let $A, B \in \Sigma$ be events of $\EE$. We write the '''conditional probability of $A$ given $B$''' as $\condprob A B$, and define it as: :''the probability that $A$ has occurred, given that $B$ has occurred.''
Definition:Independent Events
Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$. Let $A, B \in \Sigma$ be events of $\EE$ such that $\map \Pr A > 0$ and $\map \Pr B > 0$. === Definition 1 === {{:Definition:Independent Events/Definition 1}} === Definition 2 === {{:Definition:Independent Events/Definition 2}}
Definition:Disjoint Events
Let $A$ and $B$ be events in a probability space. Then $A$ and $B$ are '''disjoint''' : :$A \cap B = \O$ It follows by definition of probability measure that $A$ and $B$ are '''disjoint''' : :$\map \Pr {A \cap B} = 0$
Definition:Can't Happen
=== Probability Theory === {{:Definition:Can't Happen/Probability Theory}} === Computer Science === {{:Definition:Can't Happen/Computer Science}}
Definition:Partition (Probability Theory)
Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. A '''partition of $\Omega$''' is a family $\left\{{B_i: i \in I}\right\}$ of disjoint events such that $\displaystyle \bigcup_i B_i = \Omega$.
Definition:Increasing Sequence of Events
Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. Let $\left \langle {A_n}\right \rangle$ be a sequence of events in $\Sigma$. Then $\left \langle {A_n}\right \rangle$ is described as '''increasing''' iff: :$\forall i \in \N: A_i \subseteq A_{i+1}$
Definition:Limit of Sequence of Events
Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. === Increasing Sequence of Events === {{:Definition:Limit of Sequence of Events/Increasing}} === Decreasing Sequence of Events === {{:Definition:Limit of Sequence of Events/Decreasing}}
Definition:Decreasing Sequence of Events
Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. Let $\left \langle {A_n}\right \rangle$ be a sequence of events in $\Sigma$. Then $\left \langle {A_n}\right \rangle$ is described as '''decreasing''' iff: :$\forall i \in \N: A_{i+1} \subseteq A_i$
Definition:Random Variable/Discrete
Let $\EE$ be an experiment with a probability space $\struct {\Omega, \Sigma, \Pr}$. <!-- A '''discrete random variable''' on $\struct {\Omega, \Sigma, \Pr}$ is a random variable on $\struct {\Omega, \Sigma, \Pr}$ such that $\Omega_X$, the image of $X$, is a countable subset of $\R$. --> A '''discrete random variable''' on $\struct {\Omega, \Sigma, \Pr}$ is a mapping $X: \Omega \to \R$ such that: :$(1): \quad$ The image of $X$ is a countable subset of $\R$ :$(2): \quad$ $\forall x \in \R: \set {\omega \in \Omega: \map X \omega = x} \in \Sigma$ Alternatively (and meaning exactly the same thing), the second condition can be written as: :$(2)': \quad$ $\forall x \in \R: \map {X^{-1} } x \in \Sigma$ where $\map {X^{-1} } x$ denotes the preimage of $x$. Note that if $x \in \R$ is not the image of any elementary event $\omega$, then $\map {X^{-1} } x = \O$ and of course by definition of event space as a sigma-algebra, $\O \in \Sigma$. Note that a discrete random variable also fulfils the conditions for it to be a random variable.
Definition:Probability Mass Function
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X: \Omega \to \R$ be a discrete random variable on $\struct {\Omega, \Sigma, \Pr}$. Then the '''(probability) mass function''' of $X$ is the (real-valued) function $p_X: \R \to \closedint 0 1$ defined as: :$\forall x \in \R: \map {p_X} x = \begin{cases} \map \Pr {\set {\omega \in \Omega: \map X \omega = x} } & : x \in \Omega_X \\ 0 & : x \notin \Omega_X \end{cases}$ where $\Omega_X$ is defined as $\Img X$, the image of $X$. That is, $\map {p_X} x$ is the probability that the discrete random variable $X$ takes the value $x$.
Definition:Bernoulli Distribution
Let $X$ be a discrete random variable on a probability space. Then $X$ has the '''Bernoulli distribution with parameter $p$''' : : $(1): \quad X$ has exactly two possible values, for example $\Img X = \set {a, b}$ : $(2): \quad \map \Pr {X = a} = p$ : $(3): \quad \map \Pr {X = b} = 1 - p$ where $0 \le p \le 1$.
Definition:Bernoulli Trial
A '''Bernoulli trial''' is an experiment whose sample space has two elements, which can be variously described, for example, as: :'''Success''' and '''failure''' :'''True''' and '''False''' :$1$ and $0$ :the classic '''heads''' and '''tails'''. Formally, a '''Bernoulli trial''' is modelled by a probability space $\struct {\Omega, \Sigma, \Pr}$ such that: :$\Omega = \set {a, b}$ :$\Sigma = \powerset \Omega$ :$\map \Pr a = p, \map \Pr b = 1 - p$ where: :$\powerset \Omega$ denotes the power set of $\Omega$ :$0 \le p \le 1$ That is, $\Pr$ obeys a Bernoulli distribution. === Bernoulli Variable === {{:Definition:Bernoulli Trial/Bernoulli Variable}}
Definition:Binomial Distribution
Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Then $X$ has the '''binomial distribution with parameters $n$ and $p$''' : :$\Img X = \set {0, 1, \ldots, n}$ :$\map \Pr {X = k} = \dbinom n k p^k \paren {1 - p}^{n - k}$ where $0 \le p \le 1$.
Definition:Poisson Distribution
Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Then $X$ has the '''Poisson distribution with parameter $\lambda$''' (where $\lambda > 0$) : :$\Img X = \set {0, 1, 2, \ldots} = \N$ :$\map \Pr {X = k} = \dfrac 1 {k!} \lambda^k e^{-\lambda}$ It is written: :$X \sim \Poisson \lambda$
Definition:Geometric Distribution
Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. $X$ has the '''geometric distribution with parameter $p$''' : :$\map X \Omega = \set {0, 1, 2, \ldots} = \N$ :$\map \Pr {X = k} = \paren {1 - p} p^k$ where $0 < p < 1$.
Definition:Negative Binomial Distribution
Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. There are two forms of the '''negative binomial distribution''', as follows: === First Form === {{:Definition:Negative Binomial Distribution/First Form}} === Second Form === {{:Definition:Negative Binomial Distribution/Second Form}}
Definition:Bernoulli Process
A '''Bernoulli process''' is a sequence (either finite or infinite) of Bernoulli trials, each of which has the same parameter $p$. That is, '''Bernoulli process (with parameter $p$)''' is a sequence $\left \langle {X_i}\right \rangle$ (either finite or infinite) such that: * The value of each $X_i$ is one of two values (for example: $a$ or $b$). * The probability that $X_i = a$ is the same for all $i$ (for example: $p$). That is, it is a sequence of experiments, all of which can be modelled by the same Bernoulli distribution. Note: The assumption is that the outcomes of all the Bernoulli trials are independent. {{NamedforDef|Jacob Bernoulli|cat=Bernoulli, Jacob}}
Definition:Expectation
The '''expectation''' of a random variable is the arithmetic mean of its values.
Definition:Normalized Weight Function
Let $S = \left \langle {x_1, x_2, \ldots, x_n}\right \rangle$ be a sequence of real numbers. Let $W \left({x}\right)$ be a weight function to be applied to the elements of $S$. Then $W$ is defined as being '''normalized''' if: :$\displaystyle \sum_x W \left({x}\right) = 1$   2r5oz2hi2xbnc4etn6rnpkl1y4pooah
Definition:Variance/Discrete
Let $X$ be a discrete random variable. Then the '''variance of $X$''', written $\var X$, is a measure of how much the values of $X$ varies from the expectation $\expect X$, and is defined as:
Definition:Conditional Expectation
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X$ be a discrete random variable on $\struct {\Omega, \Sigma, \Pr}$. Let $B$ be an event in $\struct {\Omega, \Sigma, \Pr}$ such that $\map \Pr B > 0$. The '''conditional expectation of $X$ given $B$''' is written $\expect {X \mid B}$ and defined as: :$\expect {X \mid B} = \displaystyle \sum_{x \mathop \in \image X} x \, \map \Pr {X = x \mid B}$ where: :$\map \Pr {X = x \mid B}$ denotes the conditional probability that $X = x$ given $B$ whenever this sum converges absolutely.
Definition:Independent Random Variables
Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$. Let $X$ and $Y$ be random variables on $\struct {\Omega, \Sigma, \Pr}$. Then $X$ and $Y$ are defined as '''independent (of each other)''' : :$\map \Pr {X = x, Y = y} = \map \Pr {X = x} \map \Pr {Y = y}$ where $\map \Pr {X = x, Y = y}$ is the joint probability mass function of $X$ and $Y$.
Definition:Uniform Distribution/Discrete
Let $X$ be a discrete random variable on a probability space. Then $X$ has a '''discrete uniform distribution with parameter $n$''' : :$\Img X = \set {1, 2, \ldots, n}$ :$\map \Pr {X = k} = \dfrac 1 n$ That is, there is a number of outcomes with an equal probability of occurrence.  This is written: :$X \sim \DiscreteUniform n$
Definition:Cumulative Distribution Function
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$. The '''cumulative distribution function''' (or '''c.d.f.''') of $X$ is denoted $\map F X$, and defined as: :$\forall x \in \R: \map {\map F X} x := \map \Pr {X \le x}$
Definition:Random Variable/Continuous
Let $\mathcal E$ be an experiment with a probability space $\struct {\Omega, \Sigma, \Pr}$. A '''continuous random variable''' on $\struct {\Omega, \Sigma, \Pr}$ is a random variable $X: \Omega \to \R$ whose cumulative distribution function is continuous for all $x \in \R$.
Definition:Degenerate Distribution
Let $X$ be a discrete random variable on a probability space. Then $X$ has a '''degenerate distribution with parameter $r$''' if: :$\Omega_X = \set r$ :$\map \Pr {X = k} = \begin{cases} 1 & : k = r \\ 0 & : k \ne r \end{cases}$ That is, there is only value that $X$ can take, namely $r$, which it takes with certainty.
Definition:Probability Generating Function
Let $X$ be a discrete random variable whose codomain, $\Omega_X$, is a subset of the natural numbers $\N$. Let $p_X$ be the probability mass function for $X$. The '''probability generating function for $X$''', denoted $\map {\Pi_X} s$, is the formal power series defined by: :$\displaystyle \map {\Pi_X} s := \sum_{n \mathop = 0}^\infty \map {p_X} n s^n \in \R \left[\left[{s}\right]\right]$
Definition:Convergent Series
Let $\left({S, \circ, \tau}\right)$ be a topological semigroup. Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a series in $S$. This series is said to be '''convergent'''  its sequence of partial sums $\left \langle {s_N} \right \rangle$ converges in the topological space $\left({S, \tau}\right)$. If $s_N \to s$ as $N \to \infty$, the series '''converges to the sum $s$''', and one writes $\displaystyle \sum_{n \mathop = 1}^\infty a_n = s$. === Convergent Series in a Normed Vector Space (Definition 1) === {{:Definition:Convergent Series/Normed Vector Space/Definition 1}} === Convergent Series in a Normed Vector Space (Definition 2) === {{:Definition:Convergent Series/Normed Vector Space/Definition 2}} === Convergent Series in a Number Field === {{:Definition:Convergent Series/Number Field}}
Definition:Probability Distribution
Let $\tuple {\Omega, \Sigma, \Pr}$ be a probability space. Let $X: \Omega \to \R$ be a random variable on $\tuple {\Omega, \Sigma, \Pr}$. Then the '''probability distribution of $X$''' is the pushforward $X_* \Pr$ of $\Pr$ on $\tuple {\R, \map \BB \R}$, where $\map \BB \R$ denotes the Borel $\sigma$-algebra on $\R$.