hcju/proofwikips huggingface dataset:
Filter on Product Space Converges to Point iff Projections Converge to Projections of Point
Let $\family{X_i}_{i \mathop \in I}$ be an indexed family of non-empty topological spaces where $I$ is an arbitrary index set. Let $\displaystyle X := \prod_{i \mathop \in I} X_i$ be the corresponding product space. Let $\FF$ be a filter on $X$. Let $x \in X$. Let $\pr_i: X \to X_i$ denote the projection from $X$ onto $X_i$. Then $\FF$ converges to $x$ : :for all $i \in I$ the image filter $\map {\pr_i} \FF$ converges to $x_i := \map {\pr_i} x$.
Image of Ultrafilter is Ultrafilter
Let $X, Y$ be two sets, $f: X \to Y$ a mapping and $\mathcal F$ an ultrafilter on $X$. Then the image filter $f \left({\mathcal F}\right)$ is an ultrafilter on $Y$.
Schur's Inequality
Let $x, y, z \in \R_{\ge 0}$ be positive real numbers. Let $t \in \R, t > 0$ be a (strictly) positive real number. Then: :$x^t \paren {x - y} \paren {x - z} + y^t \paren {y - z} \paren {y - x} + z^t \paren {z - x} \paren {z - y} \ge 0$ The equality holds  either: : $x = y = z$ : Two of them are equal and the other is zero. When $t$ is a positive even integer, the inequality holds for ''all'' real numbers $x, y, z$.
Wallis's Product
{{begin-eqn}} {{eqn | l = \prod_{n \mathop = 1}^\infty \frac {2 n} {2 n - 1} \cdot \frac {2 n} {2 n + 1}       | r = \frac 2 1 \cdot \frac 2 3 \cdot \frac 4 3 \cdot \frac 4 5 \cdot \frac 6 5 \cdot \frac 6 7 \cdot \frac 8 7 \cdot \frac 8 9 \cdots       | c =  }} {{eqn | r = \frac \pi 2       | c =  }} {{end-eqn}}
Number of Regions in Plane Defined by Given Number of Lines
The maximum number $L_n$ of regions in the plane that can be defined by $n$ straight lines in the plane is: :$L_n = \dfrac {n \paren {n + 1} } 2 + 1$ {{OEIS|A000124}}
Forward Difference of Falling Factorial
Let $f: \R \to \R$ be a real function. Let $\Delta$ denote the forward difference operator. Let $x^{\underline m}$ be the $m$th falling factorial of $x$ Then: :$\map \Delta {x^{\underline m} } = m x^{\underline {m - 1} }$
Chen's Theorem
Every sufficiently large even integer is the sum of either: :$(1): \quad$ two primes, or  :$(2): \quad$ a prime and a semiprime.
Binomial Coefficient involving Prime
Let $p$ be a prime number. Let $\dbinom n p$ be a binomial coefficient. Then: :$\dbinom n p \equiv \floor {\dfrac n p} \pmod p$ where: :$\floor {\dfrac n p}$ denotes the floor function.
Binomial Coefficient of Prime Minus One Modulo Prime
Let $p$ be a prime number. Then: :$0 \le k \le p - 1 \implies \dbinom {p - 1} k \equiv \left({-1}\right)^k \pmod p$ where $\dbinom {p - 1} k$ denotes a binomial coefficient.
Number of Primes is Infinite
The number of primes is infinite.
Quotient and Remainder to Number Base
Let $n \in \Z: n > 0$ be an integer. Let $n$ be expressed in base $b$: :$\displaystyle n = \sum_{j \mathop = 0}^m {r_j b^j}$ i.e. :$n = \left[{r_m r_{m-1} \ldots r_2 r_1 r_0}\right]_b$ Then: : $\displaystyle \left \lfloor {\frac n b} \right \rfloor = \left[{r_m r_{m-1} \ldots r_2 r_1}\right]_b$ : $n \bmod b = r_0$ where: : $\left \lfloor {.} \right \rfloor$ denotes the floor function; : $n \bmod b$ denotes the modulo operation.
Factorial Divisible by Prime Power
Let $n \in \Z: n \ge 1$. Let $p$ be a prime number. Let $n$ be expressed in base $p$ notation: :$\displaystyle n = \sum_{j \mathop = 0}^m r_j p^j$ where $0 \le r_j < p$. Let $n!$ be the factorial of $n$. Let $p^\mu$ be the largest power of $p$ which divides $n!$, that is: :$p^\mu \divides n!$ :$p^{\mu + 1} \nmid n!$ Then: :$\mu = \dfrac {n - \map {s_p} n} {p - 1}$ where $\map {s_p} n$ is the digit sum of $n$.
Wilson's Theorem/Corollary 2
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $p$ be a prime factor of $n!$ with multiplicity $\mu$. Let $n$ be expressed in a base $p$ representation as: {{begin-eqn}} {{eqn | l = n       | r = \sum_{j \mathop = 0}^m a_j p^j       | c = where $0 \le a_j < p$ }} {{eqn | r = a_0 + a_1 p + a_2 p^2 + \cdots + a_m p^m       | c = for some $m > 0$ }} {{end-eqn}} Then: :$\dfrac {n!} {p^\mu} \equiv \paren {-1}^\mu a_0! a_1! \dotsb a_m! \pmod p$
Negative Number is Congruent to Modulus minus Number
:$\forall m, n \in \Z: -m \equiv n - m \pmod n$ where $\bmod n$ denotes congruence modulo $n$.
Gaussian Integers form Subring of Complex Numbers
The ring of Gaussian integers: :$\struct {\Z \sqbrk i, +, \times}$ forms a subring of the set of complex numbers $\C$.
Gaussian Rationals form Subfield of Complex Numbers
The set of Gaussian rationals $\Q \left[{i}\right]$, under the operations of complex addition and complex multiplication, forms a subfield of the set of complex numbers $\C$.
Complex Conjugation is Automorphism
Consider the field of complex numbers $\C$. The operation of complex conjugation: :$\forall z \in \C: z \mapsto \overline z$ is a field automorphism.
Tschirnhaus Transformation
Let $P_n \left({x}\right) = 0$ be a polynomial equation of order $n$: :$a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 = 0$ Then the substitution: $y = x + \dfrac {a_{n-1}} {n a_n}$ converts $P_n$ into a depressed polynomial: :$b_n y^n + b_{n-1} y^{n-1} + \cdots + b_1 y + b_0 = 0$ where $b_{n-1} = 0$. Such a substitution is called a '''Tschirnhaus transformation'''.
Elementary Properties of Probability Measure
Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$. The probability measure $\Pr$ of $\EE$ has the following properties:
Elementary Properties of Event Space
Let $\mathcal E$ be an experiment with a probability space $\left({\Omega, \Sigma, \Pr}\right)$. The event space $\Sigma$ of $\mathcal E$ has the following properties:
Ring of Sets is Semiring of Sets
Let $\mathcal R$ be a ring of sets. Then $\mathcal R$ is also a semiring of sets.
Set of All Real Intervals is Semiring of Sets
Let $\mathbb S$ be the set of all real intervals. Then $\mathbb S$ is a semiring of sets, but is '''not''' a ring of sets.
Ring of Sets Generated by Semiring
Let $\mathcal S$ be a semiring of sets. Let $\mathcal R \left({\mathcal S}\right)$ be the minimal ring generated by $\mathcal S$. Let $\mathcal L$ be the system of sets $A$ with the finite expansions: :$\displaystyle A = \bigcup_{k \mathop = 1}^n A_k$ with respect to the sets $A_k \in \mathcal S$. Then $\mathcal L = \mathcal R \left({\mathcal S}\right)$.
Ring of Sets Closed under Finite Intersection
Let $\mathcal R$ be a ring of sets. Let $A_1, A_2, \ldots, A_n \in \mathcal R$. Then: :$\displaystyle \bigcap_{j \mathop = 1}^n A_j \in \mathcal R$
Sigma-Algebra is Delta-Algebra
A $\sigma$-algebra is also a $\delta$-algebra.
Addition Law of Probability
:$\map \Pr {A \cup B} = \map \Pr A + \map \Pr B - \map \Pr {A \cap B}$
Cardinality is Additive Function
Let $S$ be a finite set. Let $\mathcal P \left({S}\right)$ be the power set of $S$. The function $C: \mathcal P \left({S}\right) \to \R$, where $C$ is defined as the cardinality of a set, is an additive function.
Finite Union of Sets in Additive Function
Let $\mathcal A$ be an algebra of sets. Let $f: \mathcal A \to \overline {\R}$ be an additive function. Let $A_1, A_2, \ldots, A_n$ be any finite collection of pairwise disjoint elements of $\mathcal A$. Then: :$\displaystyle f \left({\bigcup_{i \mathop = 1}^n A_i}\right) = \sum_{i \mathop = 1}^n f \left({A_i}\right)$ That is, for any collection of pairwise disjoint elements of $\mathcal A$, $f$ of their union equals the sum of $f$ of the individual elements.
Measure is Finitely Additive Function
Let $\Sigma$ be a $\sigma$-algebra on a set $X$. Let $\mu: \Sigma \to \overline {\R}$ be a measure on $\Sigma$. Then $\mu$ is finitely additive.
Finite Union of Sets in Subadditive Function
Let $\mathcal A$ be an algebra of sets. Let $f: \mathcal A \to \overline {\R}$ be a subadditive function. Let $A_1, A_2, \ldots, A_n$ be any finite collection of elements of $\mathcal A$. Then: :$\displaystyle f \left({\bigcup_{i \mathop = 1}^n A_i}\right) \le \sum_{i \mathop = 1}^n f \left({A_i}\right)$ That is, for any finite collection of elements of $\mathcal A$, $f$ of their union is less than or equal to the sum of $f$ of the individual elements.
Additive Function on Empty Set is Zero
Let $\mathcal A$ be an algebra of sets. Let $f: \mathcal A \to \overline {\R}$ be an additive function on $\mathcal A$. Then $f \left({\varnothing}\right) = 0$.
Additive Function is Strongly Additive
Let $\SS$ be an algebra of sets. Let $f: \SS \to \overline \R$ be an additive function on $\SS$. Then $f$ is also strongly additive. That is: :$\forall A, B \in \SS: \map f {A \cup B} + \map f {A \cap B} = \map f A + \map f B$
Set Difference is Disjoint with Reverse
:$\paren {S \setminus T} \cap \paren {T \setminus S} = \O$
Measure is Monotone
Let $\struct {X, \Sigma, \mu}$ be a measure space. Then $\mu$ is monotone, that is: :$\forall E, F \in \Sigma: E \subseteq F \implies \map \mu E \le \map \mu F$
Countably Additive Function also Finitely Additive
Let $\mathcal A$ be a $\sigma$-algebra. Let $f: \mathcal A \to \overline \R$ be a function, where $\overline \R$ denotes the set of extended real numbers. Let $f$ be a countably additive function: :$\displaystyle f \left({\bigcup_{i \mathop \in \N} A_i}\right) = \sum_{i \mathop \in \N} f \left({A_i}\right)$ such that there exists at least one $A \in \mathcal A$ such that $f \left({A}\right)$ is a finite number. Then $f$ is a finitely additive function.
Countably Additive Function of Null Set
Let $\mathcal A$ be a $\sigma$-algebra. Let $f: \mathcal A \to \overline \R$ be a function, where $\overline \R$ denotes the extended set of real numbers. Let $f$ be a countably additive function: :$\displaystyle f \left({\bigcup_{i \mathop \in \N} A_i}\right) = \sum_{i \mathop \in \N} f \left({A_i}\right)$ such that there exists at least one $A \in \mathcal A$ where $f \left({A}\right)$ is a finite number. Then: : $f \left({\varnothing}\right) = 0$
Non-Negative Additive Function is Monotone
Let $\SS$ be an algebra of sets. Let $f: \SS \to \overline \R$ be an additive function, that is: :$\forall A, B \in \SS: A \cap B = \O \implies \map f {A \cup B} = \map f A + \map f B$ If $\forall A \in \SS: \map f A \ge 0$, then $f$ is monotone, that is: :$A \subseteq B \implies \map f A \le \map f B$
Power Set is Sigma-Algebra
The power set of a set is a sigma-algebra.
Probability Measure on Equiprobable Outcomes
Let $\struct {\Omega, \Sigma, \Pr}$ be an equiprobability space. Let $\card \Omega = n$. Then: :$\forall \omega \in \Omega: \map \Pr \omega = \dfrac 1 n$ :$\forall A \subseteq \Omega: \map \Pr A = \dfrac {\card A} n$.
Conditional Probability Defines Probability Space
Let $\left({\Omega, \Sigma, \Pr}\right)$ be a measure space. Let $B \in \Sigma$ such that $\Pr \left({B}\right) > 0$. Let $Q: \Sigma \to \R$ be the real-valued function defined as: :$Q \left({A}\right) = \Pr \left({A \mid B}\right)$ where: :$\Pr \left({A \mid B}\right) = \dfrac {\Pr \left({A \cap B}\right)}{\Pr \left({B}\right)}$ is the conditional probability of $A$ given $B$. Then $\left({\Omega, \Sigma, Q}\right)$ is a probability space.
Bayes' Theorem
Let $\Pr$ be a probability measure on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $\map \Pr {A \mid B}$ denote the conditional probability of $A$ given $B$. Let $\map \Pr A > 0$ and $\map \Pr B > 0$. Then: :$\map \Pr {B \mid A} = \dfrac {\map \Pr {A \mid B} \, \map \Pr B} {\map \Pr A}$
Event Independence is Symmetric
Let $A$ and $B$ be events in a probability space. Let $A$ be independent of $B$. Then $B$ is independent of $A$. That is, '''is independent of''' is a symmetric relation.
Set Difference and Intersection form Partition
Let $S$ and $T$ be sets such that: :$S \setminus T \ne \O$ :$S \cap T \ne \O$ where $S \setminus T$ denotes set difference and $S \cap T$ denotes set intersection. Then $S \setminus T$ and $S \cap T$ form a partition of $S$.
Independent Events are Independent of Complement
Let $A$ and $B$ be events in a probability space $\struct {\Omega, \Sigma, \Pr}$. Then $A$ and $B$ are independent  $A$ and $\Omega \setminus B$ are independent.
Probability of Independent Events Not Happening
Let $\EE = \struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $A_1, A_2, \ldots, A_m \in \Sigma$ be independent events in the event space of $\EE$. Then the probability of none of $A_1$ to $A_m$ occurring is: :$\displaystyle \prod_{i \mathop = 1}^m \paren {1 - \map \Pr {A_i} }$
Probability of Occurrence of At Least One Independent Event
Let $\EE = \struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $A_1, A_2, \ldots, A_m \in \Sigma$ be independent events in the event space of $\EE$. Then the probability of at least one of $A_1$ to $A_m$ occurring is: :$\displaystyle 1 - \prod_{i \mathop = 1}^m \paren {1 - \map \Pr {A_i} }$
Total Probability Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $\set {B_1, B_2, \ldots}$ be a partition of $\Omega$ such that $\forall i: \map \Pr {B_i} > 0$. Then: :$\displaystyle \forall A \in \Sigma: \map \Pr A = \sum_i \map \Pr {A \mid B_i} \, \map \Pr {B_i}$
Probability of Limit of Sequence of Events
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Boole's Inequality
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $A_1, A_2, \ldots, A_n$ be events in $\Sigma$. Then: :$\displaystyle \map \Pr {\bigcup_{i \mathop = 1}^n A_i} \le \sum_{i \mathop = 1}^n \map \Pr {A_i}$
Additive Nowhere Negative Function is Subadditive
Let $\AA$ be an algebra of sets. Let $f: \AA \to \overline {\R}$ be an additive function such that: :$\forall A \in \AA: \map f A \ge 0$ Then $f$ is subadditive.
Probability Measure is Subadditive
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Then $\Pr$ is a subadditive function.
Inclusion-Exclusion Principle
Let $\SS$ be an algebra of sets. Let $A_1, A_2, \ldots, A_n$ be finite sets. Let $f: \SS \to \R$ be an additive function. Then: {{begin-eqn}} {{eqn | l = \map f {\bigcup_{i \mathop = 1}^n A_i}       | r = \sum_{i \mathop = 1}^n \map f {A_i}       | c =  }} {{eqn | o =        | ro= -       | r = \sum_{1 \mathop \le i \mathop < j \mathop \le n} \map f {A_i \cap A_j}       | c =  }} {{eqn | o =        | ro= +       | r = \sum_{1 \mathop \le i \mathop < j \mathop < k \mathop \le n} \map f {A_i \cap A_j \cap A_k}       | c =  }} {{eqn | o =        | ro= \cdots       | c =  }} {{eqn | o =        | ro= +       | r = \paren {-1}^{n - 1} \map f {\bigcap_{i \mathop = 1}^n A_i}       | c =  }} {{end-eqn}}
Existence of Probability Space and Discrete Random Variable
Let $I$ be some indexing set. Let $S = \left\{{s_i: i \in I}\right\} \subset \R$ be a countable set of real numbers. Let $\left\{{\pi_i: i \in I}\right\} \subset \R$ be a countable set of real numbers which satisfies: :$\displaystyle \forall i \in I: \pi_i \ge 0, \sum_{i \mathop \in I} \pi_i = 1$ Then there exists a probability space $\left({\Omega, \Sigma, \Pr}\right)$ and a discrete random variable $X$ on $\left({\Omega, \Sigma, \Pr}\right)$ such that the probability mass function $p_X$ of $X$ is given by: {{begin-eqn}} {{eqn | l=p_X \left({s_i}\right)       | r=\pi_i       | c=if $i \in I$ }} {{eqn | l=p_X \left({s}\right)       | r=0       | c=if $s \notin S$ }} {{end-eqn}}
Sum and Product of Discrete Random Variables
Let $X$ and $Y$ be discrete random variables on the probability space $\left({\Omega, \Sigma, \Pr}\right)$.
Countable Function on Power Set of Sample Space is Discrete Random Variable
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space such that $\Sigma$ is the power set of $\Omega$. Let $f: \Omega \to \R$ be a function such that $\Img f$ is countable. Then $f$ is a discrete random variable on $\struct {\Omega, \Sigma, \Pr}$.
Characteristic Function on Event is Discrete Random Variable
Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. Let $E \in \Sigma$ be any event of $\left({\Omega, \Sigma, \Pr}\right)$. Let $\chi_E: \Omega \to \left\{{0, 1}\right\}$ be the characteristic function of $E$. Then $\chi_E$ is a discrete random variable on $\left({\Omega, \Sigma, \Pr}\right)$.
Poisson Distribution Gives Rise to Probability Mass Function
Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $X$ have the poisson distribution with parameter $\lambda$ (where $\lambda > 0$). Then $X$ gives rise to a probability mass function.
Geometric Distribution Gives Rise to Probability Mass Function
Let $X$ be a discrete random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $X$ have the geometric distribution with parameter $p$ (where $0 < p < 1$). Then $X$ gives rise to a probability mass function.
Negative Binomial Distribution Gives Rise to Probability Mass Function/Second Form
Let $X$ have the negative binomial distribution (second form) with parameters $n$ and $p$ ($0 < p < 1$). Then $X$ gives rise to a probability mass function.
Bernoulli Process as Binomial Distribution
Let $\left \langle{X_i}\right \rangle$ be a finite Bernoulli process of length $n$ such that each of the $X_i$ in the sequence is a Bernoulli trial with parameter $p$. Then the number of successes in $\left \langle{X_i}\right \rangle$ is modelled by a binomial distribution with parameters $n$ and $p$. Hence it can be seen that: :$X \sim \operatorname{B} \left({1, p}\right)$ is the same thing as $X \sim \operatorname{Bern} \left({p}\right)$
Binomial Distribution Approximated by Poisson Distribution
Let $X$ be a discrete random variable which has the binomial distribution with parameters $n$ and $p$. Suppose $n$ is "very large" and $p$ is "very small", but $np$ of a "reasonable size". Then $X$ can be approximated by a Poisson distribution with parameter $\lambda$ where $\lambda = np$.
Bernoulli Process as Geometric Distribution
Let $\left \langle{X_i}\right \rangle$ be a Bernoulli process with parameter $p$. Let $\mathcal E$ be the experiment which consists of performing the Bernoulli trial $X_i$ until a failure occurs, and then stop. Let $k$ be the number of successes before a failure is encountered. Then $k$ is modelled by a geometric distribution with parameter $p$.
Function of Discrete Random Variable
Let $X$ be a discrete random variable on the probability space $\struct {\Omega, \Sigma, \Pr}$. Let $g: \R \to \R$ be any real function. Then $Y = g \sqbrk X$, defined as: :$\forall \omega \in \Omega: \map Y \omega = g \sqbrk {\map X \omega}$ is also a discrete random variable.
Expectation of Function of Discrete Random Variable
Let $X$ be a discrete random variable. Let $\expect X$ be the expectation of $X$. Let $g: \R \to \R$ be a real function. Then: :$\displaystyle \expect {g \sqbrk X} = \sum_{x \mathop \in \Omega_X} \map g x \, \map \Pr {X = x}$ whenever the sum is absolutely convergent.
Probability Mass Function of Function of Discrete Random Variable
Let $X$ be a discrete random variable. Let $Y = g \left({X}\right)$, where $g: \R \to \R$ is a real function. Then the probability mass function of $Y$ is given by: :$\displaystyle p_Y \left({y}\right) = \sum_{x \mathop \in g^{-1} \left({y}\right)} \Pr \left({X = x}\right)$
Variance as Expectation of Square minus Square of Expectation
Let $X$ be a random variable. Then the variance of $X$ can be expressed as: :$\var X = \expect {X^2} - \paren {\expect X}^2$ That is, it is the expectation of the square of $X$ minus the square of the expectation of $X$.
Derivative of Geometric Sequence
Let $x \in \R: \size x < 1$. Then: :$\displaystyle \sum_{n \mathop \ge 1} n x^{n - 1} = \frac 1 {\paren {1 - x}^2}$
Expectation of Shifted Geometric Distribution
Let $X$ be a discrete random variable with the shifted geometric distribution with parameter $p$. Then the expectation of $X$ is given by: :$\expect X = \dfrac 1 p$
Expectation of Bernoulli Distribution
Let $X$ be a discrete random variable with a Bernoulli distribution with parameter $p$. Then the expectation of $X$ is given by: :$\expect X = p$
Expectation of Binomial Distribution
Let $X$ be a discrete random variable with the binomial distribution with parameters $n$ and $p$ for some $n \in \N$ and $0 \le p \le 1$. Then the expectation of $X$ is given by: :$\expect X = n p$
Expectation of Poisson Distribution
Let $X$ be a discrete random variable with the Poisson distribution with parameter $\lambda$. Then the expectation of $X$ is given by: :$\expect X = \lambda$
Variance of Bernoulli Distribution
Let $X$ be a discrete random variable with the Bernoulli distribution with parameter $p$: :$X \sim \Bernoulli p$ Then the variance of $X$ is given by: :$\var X = p \paren {1 - p}$
Variance of Poisson Distribution
Let $X$ be a discrete random variable with the Poisson distribution with parameter $\lambda$. Then the variance of $X$ is given by: :$\var X = \lambda$
Variance of Binomial Distribution
Let $X$ be a discrete random variable with the binomial distribution with parameters $n$ and $p$. Then the variance of $X$ is given by: :$\var X = n p \paren {1 - p}$
Sum of Expectations of Independent Trials
Let $\EE_1, \EE_2, \ldots, \EE_n$ be a sequence of experiments whose outcomes are independent of each other. Let $X_1, X_2, \ldots, X_n$ be discrete random variables on $\EE_1, \EE_2, \ldots, \EE_n$ respectively. Let $\expect {X_j}$ denote the expectation of $X_j$ for $j \in \set {1, 2, \ldots, n}$. Then we have, whenever both sides are defined: :$\displaystyle \expect {\sum_{j \mathop = 1}^n X_j} = \sum_{j \mathop = 1}^n \expect {X_j}$ That is, the sum of the expectations equals the expectation of the sum.
Sum of Variances of Independent Trials
Let $\EE_1, \EE_2, \ldots, \EE_n$ be a sequence of experiments whose outcomes are independent of each other. Let $X_1, X_2, \ldots, X_n$ be discrete random variables on $\EE_1, \EE_2, \ldots, \EE_n$ respectively. Let $\var {X_j}$ be the variance of $X_j$ for $j \in \set {1, 2, \ldots, n}$. Then: :$\displaystyle \var {\sum_{j \mathop = 1}^n X_j} = \sum_{j \mathop = 1}^n \var {X_j}$ That is, the sum of the variances equals the variance of the sum.
Variance of Shifted Geometric Distribution
Let $X$ be a discrete random variable with the shifted geometric distribution with parameter $p$. Then the variance of $X$ is given by: :$\var X = \dfrac {1 - p} {p^2}$
Expectation of Geometric Distribution
Let $X$ be a discrete random variable with the geometric distribution with parameter $p$. Then the expectation of $X$ is given by: :$E \left({X}\right) = \dfrac p {1-p}$
Variance of Geometric Distribution
Let $X$ be a discrete random variable with the geometric distribution with parameter $p$. Then the variance of $X$ is given by: :$\operatorname{var} \left({X}\right) = \dfrac p {\left({1 - p}\right)^2}$
Bernoulli Process as Negative Binomial Distribution
Let $\sequence {X_i}$ be a Bernoulli process with parameter $p$.
Properties of Cumulative Distribution Function
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$. Let $\map F X$ be the cumulative distribution function of $X$: :$\forall x \in \R: \map F x = \map \Pr {X \le x}$ Then the following conditions apply to $\map F X$:
Discrete Random Variable is Random Variable
Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space. Let $X$ be a discrete random variable on $\left({\Omega, \Sigma, \Pr}\right)$. Then $X$ fulfils the condition: :$\forall x \in \R: \left\{{\omega \in \Omega: X \left({\omega}\right) \le x}\right\} \in \Sigma$ That is, $X$ fulfils the condition for it to be a random variable.
Expectation of Discrete Uniform Distribution
Let $X$ be a discrete random variable with the discrete uniform distribution with parameter $n$. Then the expectation of $X$ is given by: :$\expect X = \dfrac {n + 1} 2$
Variance of Discrete Uniform Distribution
Let $X$ be a discrete random variable with the discrete uniform distribution with parameter $p$. Then the variance of $X$ is given by: :$\var X = \dfrac {n^2 - 1} {12}$
Probability Generating Function of Degenerate Distribution
Let $X$ be the degenerate distribution: :$\forall x \in \N: p_X \left({x}\right) = \begin{cases} 1 & : x = k \\ 0 & : x \ne k \end{cases}$ where $k \in \N$. Then the p.g.f. of $X$ is: :$\Pi_X \left({s}\right) = s^k$
Probability Generating Function of Bernoulli Distribution
Let $X$ be a discrete random variable with the Bernoulli distribution with parameter $p$. Then the p.g.f. of $X$ is: :$\Pi_X \left({s}\right) = q + ps$ where $q = 1 - p$.