nyuuzyou/wikis huggingface dataset:
Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit

== Theorem ==
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a$ be a $p$-adic number, that is left coset, in $\Q_p$.
Let $\sequence{x_n}$ be a rational sequence.
Then:
:$\sequence{x_n}$ converges to $a$ $\sequence{x_n}$ is a representative of $a$
=== Corollary ===
== Proof ==
Let $\norm {\,\cdot\,}^\Q_p$ be the p-adic norm on the rationals $\Q$.
By definition of the $p$-adic numbers:
:$\Q_p$ is the quotient ring $\CC \, \big / \NN$
where:
:$\CC$ is the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$.
and
:$\NN$ is the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$.
By definition of the $p$-adic numbers, $\norm {\, \cdot \,}_p:\Q_p \to \R_{\ge 0}$ is the norm on the quotient ring $\Q_p$ defined by:
:$\ds \forall \sequence {x_n} + \NN: \norm {\sequence {x_n} + \NN }_p = \lim_{n \mathop \to \infty} \norm{x_n}_p$
From Rational Numbers are Dense Subfield of P-adic Numbers, the mapping $\phi: \Q \to \Q_p$ defined by:
:$\map \phi r = \tuple {r, r, r, \dotsc} + \NN$
::where $\tuple {r, r, r, \dotsc}$ is the constant sequence
embeds $\Q$ as a dense subfield of $\Q_p$.
The theorem follows immediately from Normed Division Ring Sequence Converges in Completion iff Sequence Represents Limit.
Category:P-adic Number Theory
Category:Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit
Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit/P-adic Expansion

== Theorem ==
Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $a$ be a $p$-adic number, that is left coset, in $\Q_p$.
Let $\ds \sum_{n \mathop = m}^\infty d_n p^n$ be a $p$-adic expansion.
Then:
:$\ds \sum_{n \mathop = m}^\infty d_n p^n$ converges to $a$ $\ds \sum_{n \mathop = m}^\infty d_n p^n$ is a representative of $a$
== Proof ==
By definition of a $p$-adic expansion:
:$\ds \sum_{n \mathop = m}^\infty d_n p^n$ is a rational sequence.
The theorem follows from Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit.
Category:Rational Sequence Converges in P-adic Numbers iff Sequence Represents Limit
Rational Sequence Decreasing to Real Number

== Theorem ==
Let $x \in \R$ be a real number.
Then there exists some decreasing rational sequence that converges to $x$.
== Proof ==
Let $\sequence {x_n}$ denote the sequence defined as:
:$\forall n \in \N : x_n = \dfrac {\ceiling {n x} } n$
where $\ceiling {n x}$ denotes the ceiling of $n x$.
From Ceiling Function is Integer, $\ceiling {n x}$ is an integer.
Hence by definition of rational number, $\sequence {x_n}$ is a rational sequence.
From Real Number is between Ceiling Functions:
:$n x < \ceiling {n x} \le n x + 1$
Thus:
:$x < \dfrac {\ceiling {n x} } n \le \dfrac {n x + 1} n$
Further:
Thus, from the Squeeze Theorem for Real Sequences:
:$\ds \lim_{n \mathop \to \infty} \frac {\ceiling {n x} } n = x$
From Peak Point Lemma, there is a monotone subsequence $\sequence {x_{n_k} }$ of $\sequence {x_n}$.
We have that $\sequence {x_n}$ is bounded below by $x$.
Hence $\sequence {x_{n_k} }$ is decreasing.
From Limit of Subsequence equals Limit of Real Sequence, $\sequence {x_{n_k} }$ converges to $x$.
Hence the result.
Category:Limits of Sequences
Category:Rational Sequences
Rational Sequence Increasing to Real Number

== Theorem ==
Let $x \in \R$ be a real number.
Then there exists some increasing rational sequence that converges to $x$.
== Proof ==
Let $\sequence {x_n}$ denote the sequence defined as:
:$\forall n \in \N: x_n = \dfrac {\floor {n x} } n$
where $\floor {n x}$ denotes the floor of $n x$.
From Floor Function is Integer, $\floor {n x}$ is an integer.
Hence by definition of rational number, $\sequence {x_n}$ is a rational sequence.
From Real Number is between Floor Functions:
:$n x - 1 < \floor {n x} \le n x$
Thus:
:$\dfrac {n x - 1} n < \dfrac {\floor {n x} } n \le x$
Further:
From the Squeeze Theorem for Real Sequences:
:$\ds \lim_{n \mathop \to \infty} \frac {\floor {n x} } n = x$
From Peak Point Lemma, there is a monotone subsequence $\sequence {x_{n_k} }$ of $\sequence {x_n}$.
We have that $\sequence {x_n}$ is bounded above by $x$.
Hence $\sequence {x_{n_k} }$ is increasing.
From Limit of Subsequence equals Limit of Real Sequence, $\sequence {x_{n_k} }$ converges to $x$.
Hence the result.
Category:Limits of Sequences
Category:Rational Sequences
Rational Square Root of Integer is Integer

== Theorem ==
Let $n$ be an integer.
Suppose that $\sqrt n$ is a rational number.
Then $\sqrt n$ is an integer.
== Proof ==
Suppose that $\sqrt n = \dfrac a b$, with $a, b$ coprime integers and $b > 0$.
Then we would have:
:$n = \dfrac {a^2} {b^2}$
That is:
:$n b^2 = a^2$
From Number divides Number iff Square divides Square:
:$b^2 \divides a^2 \implies b \divides a$
However, since $a \perp b$ and $b \divides a$, this means that necessarily $b = 1$.
That is, $\sqrt n = a$, an integer.
Hence the result.
== Sources ==
* : Chapter $2$: Some Properties of $\Z$: Exercise $2.14$
Category:Number Theory
Category:Integers
Category:Rational Numbers
Rational Subtraction is Closed

== Theorem ==
The set of rational numbers is closed under subtraction:
:$\forall a, b \in \Q: a - b \in \Q$
== Proof ==
From the definition of subtraction:
:$a - b := a + \paren {-b}$
where $-b$ is the inverse for rational number addition.
From Rational Numbers under Addition form Infinite Abelian Group, $\struct {\Q, +}$ forms a group.
Thus:
:$\forall a, b \in \Q: a + \paren {-b} \in \Q$
Therefore rational number subtraction is closed.
== Sources ==
* : Chapter $1$: The Real and Complex Number Systems: Introduction
* : Chapter $1$: Integral Domains: $\S 2$. Operations: Example $1$
* : Chapter $1$: Real Numbers and Functions of a Real Variable: $1.1$ Real Numbers
Category:Rational Subtraction
Category:Algebraic Closure
Rationality of Rectangle Contained by Medial Straight Lines Commensurable in Square

== Theorem ==
== Proof ==
Let $A$ and $B$ be two medial straight lines.
Let $A$ and $B$ be commensurable in square only.
Then their lengths are of the form:
:$l \left({A}\right) = \rho \sqrt [4] k$
:$l \left({B}\right) = \rho \sqrt [4] k \sqrt \lambda$
where: 
: $\rho$ is a rational number
: $k$ and $\lambda$ are rational numbers whose square root is irrational.
The rectangle contained by $A$ and $B$ has area $R$ given by:
:$R = \rho^2 \sqrt k \sqrt \lambda$
In general, $R$ is medial.
However, let $\sqrt \lambda = k' \sqrt k$ where $k$ is rational.
Then:
:$R = \rho^2 \sqrt k k' \sqrt k = \rho^2 k k'$
which is rational.
== Sources ==
* : Book $\text{X}$. Propositions
Category:Euclidean Number Theory
Rationalization/Examples/Arbitrary Example 2

== Example of Rationalization ==
The denominator of the irrational number:
:$\dfrac 1 {\sqrt 5 + 1}$
can be rationalized by multiplying top and bottom by $\sqrt 5 - 1$ to obtain:
:$\dfrac {\sqrt 5 - 1} 2$
== Sources ==
* 
* 
Category:Examples of Rationalization
Rationals are Dense in Compact Complement Topology

== Theorem ==
Let $T = \struct {\R, \tau^*}$ be the compact complement topology on $\R$.
Let $\Q$ be the set of rational numbers.
Then $\Q$ is everywhere dense in $T$.
== Proof ==
We have that the Compact Complement Topology is Coarser than Euclidean Topology.
The result follows from Denseness Preserved in Coarser Topology.
== Sources ==
* : Part $\text {II}$: Counterexamples: $22$. Compact Complement Topology: $6$
Category:Rational Number Space
Category:Compact Complement Topology
Category:Denseness
Rationals are Everywhere Dense in Reals/Normed Vector Space

== Theorem ==
Let $\struct {\R, \size {\, \cdot \,}}$ be the normed vector space of real numbers.
Let $\Q$ be the set of rational numbers.
Then $\Q$ are everywhere dense in $\struct {\R, \size {\, \cdot \,}}$
== Proof ==
We have that Between two Real Numbers exists Rational Number:
:$\forall a, b \in \R : a < b : \exists r \in \Q : a < r < b$
Let $a := x$ with $x \in \R$.
Let $\epsilon \in \R_{\mathop > 0} : r - a < \epsilon$.
Let $b := x + \epsilon$.
Then:
Since this holds for all $x$, we have that:
:$\forall x \in \R : \forall \epsilon \in \R_{\mathop > 0} : \exists r \in \Q : \size {x - r} < \epsilon$
By definition, $\Q$ is dense in $\R$.
== Sources ==
* : $\S 1.3$: Normed and Banach spaces. Topology of normed spaces
Category:Real Analysis
Category:Rational Number Space
Category:Denseness
Rationals are Everywhere Dense in Reals/Topology

== Theorem ==
Let $\struct {\R, \tau_d}$ denote the real number line with the usual (Euclidean) topology.
Let $\Q$ be the set of rational numbers.
Then $\Q$ is everywhere dense in $\struct {\R, \tau_d}$.
== Proof ==
Let $x \in \R$.
Let $U \subseteq \R$ be an open set of $\struct {\R, \tau_d}$ such that $x \in U$.
From Basis for Euclidean Topology on Real Number Line, there exists an open interval $V = \openint {x - \epsilon} {x + \epsilon} \subseteq U$ for some $\epsilon > 0$ such that $x \in V$.
Now consider the open interval $\openint x {x + \epsilon} \subseteq V$.
By Subset Relation is Transitive it follows that $\openint x {x + \epsilon} \subseteq U$.
Note that $x \notin \openint x {x + \epsilon}$.
From Between two Real Numbers exists Rational Number, there exists $y \in \Q: y \in \openint x {x + \epsilon}$.
As $x \notin \openint x {x + \epsilon}$, it must be the case that $x \ne y$.
That is, $V$ is an open set of $\struct {\R, \tau_d}$ containing $x$ which also contains an element of $\Q$ other than $x$.
As $V$ is arbitrary, it follows that every open set of $\struct {\R, \tau_d}$ containing $x$ also contains an element of $\Q$ other than $x$.
That is, $x$ is by definition a limit point of $\Q$.
As $x$ is arbitrary, it follows that all elements of $\R$ are limit points of $\Q$.
The result follows from the definition of everywhere dense.
== Sources ==
* : $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Definition $3.7.21$
* 
* 
Category:Real Analysis
Category:Rational Number Space
Category:Real Number Line with Euclidean Topology
Category:Denseness
Rationals are Everywhere Dense in Sorgenfrey Line

== Theorem ==
$\Q$ is everywhere dense in the Sorgenfrey line.
== Proof ==
Let $T = \struct {\R, \tau}$ be the Sorgenfrey line.
Define:
:$\BB := \set {\hointr x y: x, y \in \R}$
where $\hointr x y$ denotes the right half-open real interval between $x$ and $y$.
By definition of Sorgenfrey line:
:$\BB$ is basis of $T$.
By definition of subset:
:$\Q^- \subseteq \R$
where $\Q^-$ denotes the topological closure of $\Q$ in $T$.
By definition of set equality to prove the equality: $\Q^- = \R$, it is necessary to show:
:$\R \subseteq \Q^-$
Let $x \in \R$.
By Characterization of Closure by Basis it suffices to prove that:
:$\forall U \in \BB: x \in U \implies U \cap \Q \ne \O$
Let $U \in \BB$.
By definition of $\BB$:
:$\exists y, z \in \R: U = \hointr y z$
Assume:
:$x \in U$
By definition of half-open real interval:
:$y \le x < z$
By Between two Real Numbers exists Rational Number:
:$\exists q \in \Q: y < q < z$
By definition of half-open real interval:
:$q \in U$
Thus by definitions of intersection and empty set:
:$U \cap \Q \ne \O$
== Sources ==
* 
* 
Category:Sorgenfrey Line
Rationals plus Irrational are Everywhere Dense in Irrationals

== Theorem ==
Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$.
Let $x \in \R \setminus \Q$ be an arbitrary irrational number.
Let $S_x$ be the set defined as:
:$S_x := \set {x + q: q \in \Q}$
Then $S_x$ is everywhere dense in $\struct {\R \setminus \Q, \tau_d}$.
== Proof ==
Let $y \in \R \setminus \Q$.
Let $U \subseteq \R \setminus \Q$ be an open set of $\struct {\R \setminus \Q, \tau_d}$ such that $x \in U$.
From Basis for Euclidean Topology on Real Number Line, the set of all open real intervals of $\R$ form a basis for $\struct {\R, \tau_d}$.
By Basis for Topological Subspace, the set of all intersections of $\R \setminus \Q$ and open real intervals of $\R$ form a basis for $\struct {\R \setminus \Q, \tau_d}$.
So there exists $V = \paren {\R \setminus \Q} \cap \openint {y - \epsilon} {y + \epsilon} \subseteq U$ for some $\epsilon > 0$ such that $y \in V$.
By Between two Real Numbers exists Rational Number we have:
:$\exists r \in Q: y - \epsilon - x < r < y + \epsilon - x$.
Then:
:$y - \epsilon < x + r < y + \epsilon$
Then $x + r \in S_x \cap V \subseteq S_x \cap U$.
Since $y$ and $U$ are arbitrary:
$S_x$ is everywhere dense in $\struct {\R \setminus \Q, \tau_d}$.
== Sources ==
* : Part $\text {II}$: Counterexamples: $31$. The Irrational Numbers: $7$
Category:Irrational Number Space
Category:Denseness
Ratios of Equal Magnitudes

== Theorem ==
That is:
:$a = b \implies a : c = b : c$
:$a = b \implies c : a = c : b$
=== Porism ===
== Proof ==
Let $A, B$ be equal magnitudes and let $C$ be any other arbitrary magnitude.
We need to show that $A : C = B : C$ and $C : A = C : B$.
:
Let equimultiples $D, E$ of $A, B$ be taken, and another arbitrary multiple $F$ of $C$.
We have that $D$ is the same multiple of $A$ that $E$ is of $B$, while $A = B$.
Therefore $D = E$.
But $F$ is another arbitrary magnitude.
Therefore:
:$D > F \implies E > F$
:$D = F \implies E = F$
:$D < F \implies E < F$
We have that $D, E$ are equimultiples of $A, B$ while $F$ is another arbitrary multiple of $C$.
So from , $A : C = B : C$.
With the same construction we can show that $D = E$, while $F$ is some other magnitude.
Therefore:
:$F > D \implies F > E$
:$F = D \implies F = E$
:$F < D \implies F < E$
But $F$ is a multiple of $C$, while $D, E$ are equimultiples of $A, B$.
So from , $C : A = C : B$.
== Sources ==
* : Book $\text{V}$. Propositions
Category:Ratios
Ratios of Fractions in Lowest Terms

== Theorem ==
Let $a, b, c, d \in \Z_{>0}$ be positive integers.
Let $\dfrac a b$ be in canonical form.
Let $\dfrac a b = \dfrac c d$.
Then:
:$a \divides c$
and:
:$b \divides d$
where $\divides$ denotes divisibility.
== Proof ==
Let $CD, EF$ be the least (natural) numbers of those which have the same ratio with $A, B$.
We need to show that $CD$ measures $A$ the same number of times that $EF$ measures $B$.
:
Suppose $CD$ is an aliquant part of $A$.
Then from and , $EF$ is also the same aliquant part of $B$ that $CD$ is of $A$.
Therefore as many aliquant parts of $A$ as there are in $CD$, so many aliquant parts of $B$ are there also in $EF$.
Let $CD$ be divided into the aliquant parts of $A$, namely $CG, GD$ and $EF$ into the aliquant parts of $B$, namely $EH, HF$.
Thus the multitude of $CG, GD$ will be equal to the multitude of $EH, HF$.
We have that the numbers $CG, GD$ are equal to one another, and the numbers $EH, HF$ are also equal to one another, 
Therefore $CG : EH = GD : HF$.
So from , as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents.
Therefore $CG : EH = CD : EF$.
Therefore $CG, EH$ are in the same ratio with $CD, EF$ being less than they.
This is impossible, for by hypothesis $CD, EF$ are the least numbers of those which have the same ratio with them.
So $CD$ is not an aliquant part of $A$.
Therefore from , $CD$ is an aliquot part of $A$.
Also, from and , $EF$ is the same aliquot part of $B$ that $CD$ is of $A$.
Therefore $CD$ measures $A$ the same number of times that $EF$ measures $B$.
== Sources ==
* : Book $\text{VII}$. Propositions
Category:Ratios
Ratios of Multiples of Numbers

== Theorem ==
== Proof ==
Let two (natural) numbers $A, B$ by multiplying any number $C$ make $D, E$.
Then we need to show that:
: $A : B = D : E$
:
We have that:
: $A \times C = D$
So from Natural Number Multiplication is Commutative, also:
: $C \times A = D$
For the same reason:
: $C \times B = E$
Therefore from :
: $A : B = D : E$
== Sources ==
* : Book $\text{VII}$. Propositions
Category:Ratios
Ratios of Numbers is Distributive over Addition

== Theorem ==
== Proof ==
Let $A, B, C, D$ be as many numbers as we please in proportion, so that $A : B = C : D$.
We need to show that $A : B = A + C : B + D$.
:
We have that $A : B = C : D$.
So from , whatever aliquot part or aliquant part $A$ is of $B$, the same aliquot part or aliquant part is $C$ of $D$ also.
Therefore from:
: 
and:
: 
$A + C$ is the same aliquot part or aliquant part of $C + D$ that $A$ is of $B$.
So from , $A : B = A + C : B + D$.
== Sources ==
* : Book $\text{VII}$. Propositions
Category:Ratios
Ratios of Sizes of Mutually Inscribed Multidimensional Cubes and Spheres

== Theorem ==
Consider:
: a cube $C_n$ of $n$ dimensions inscribed within a sphere $S_n$ of $n$ dimensions
: a sphere $S'_n$ of $n$ dimensions inscribed within a cube $C'_n$ of $n$ dimensions.
Let:
: $A_{cn}$ be the $n$ dimensional volume of $C_n$
: $A_{sn}$ be the $n$ dimensional volume of $S_n$
: $A'_{cn}$ be the $n$ dimensional volume of $C'_n$
: $A'_{sn}$ be the $n$ dimensional volume of $S'_n$.
For $n < 9$:
:$\dfrac {S_n} {C_n} > \dfrac {C'_n} {S'_n}$
but for $n \ge 9$:
:$\dfrac {S_n} {C_n} < \dfrac {C'_n} {S'_n}$
That is, for dimension $n$ less than $9$, the $n$ dimensional round peg fits better into an $n$ dimensional square hole than an $n$ dimensional square peg fits into an $n$ dimensional round hole, but for $9$ and higher dimensions, the situation is reversed.
== Proof ==
== Sources ==
* 
* : $9$
* : $9$
Category:Geometry
Category:Squares
Category:Circles
Raw Moment of Bernoulli Distribution

== Theorem ==
Let $X$ be a discrete random variable with a Bernoulli distribution with parameter $p$.
Let $n$ be a strictly positive integer. 
Then the $n$th raw moment $\expect {X^n}$ of $X$ is given by: 
:$\expect {X^n} = p$
== Proof 1 ==
 
== Proof 2 ==
Category:Bernoulli Distribution
Category:Raw Moments
Category:Raw Moment of Bernoulli Distribution
Raw Moment of Bernoulli Distribution/Proof 1

== Theorem ==
== Proof ==
From the definition of expectation:
:$\ds \expect {X^n} = \sum_{x \mathop \in \Img X} x^n \map \Pr {X = x}$
From the definition of the Bernoulli distribution:
:$\ds \expect {X^n} = 1^n \times p + 0^n \times \paren {1 - p} = p$
Category:Raw Moment of Bernoulli Distribution
Raw Moment of Bernoulli Distribution/Proof 2

== Theorem ==
== Proof == 
By Moment Generating Function of Bernoulli Distribution, the moment generating function $M_X$ is given by: 
:$\map {M_X} t = q + p e^t$
By Moment in terms of Moment Generating Function:
:$\expect {X^n} = \map {M^{\paren n}_X} 0$
By Derivative of Exponential Function:
:$\map {M^{\paren n}_X} t = p e^t$
Setting $t = 0$: 
Category:Raw Moment of Bernoulli Distribution
Raw Moment of Beta Distribution

== Theorem == 
Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname {Beta}$ is the Beta distribution. 
Then:
:$\ds \expect {X^n} = \prod_{r \mathop = 0}^{n - 1} \frac {\alpha + r} {\alpha + \beta + r}$
for positive integer $n$.
== Proof == 
By Moment Generating Function of Beta Distribution, the moment generating function, $M_X$, of $X$ is given by: 
:$\ds \map {M_X} t = \expect {e^{t X} } = 1 + \sum_{n \mathop = 1}^\infty \paren {\prod_{r \mathop = 0}^{n - 1} \frac {\alpha + r} {\alpha + \beta + r} } \frac {t^n} {n!}$
We also have: 
Comparing coefficients yields the result. 
Category:Beta Distribution
Category:Raw Moments
Raw Moment of Chi-Squared Distribution

== Theorem == 
Let $n$ and $m$ be strictly positive integers. 
Let $X \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom. 
Then the $m$th raw moment $\expect {X^m}$ of $X$ is given by: 
:$\ds \expect {X^m} = \prod_{k \mathop = 0}^{m - 1} \paren {n + 2 k}$
 
== Proof == 
From the definition of the chi-squared distribution, $X$ has probability density function:
:$\ds \map {f_X} x = \dfrac 1 {2^{n / 2} \map \Gamma {n / 2} } x^{\paren {n / 2} - 1} e^{- x / 2}$
From the definition of the expected value of a continuous random variable: 
:$\ds \expect {X^m} = \int_0^\infty x^m \map {f_X} x \rd x$
So: 
Category:Raw Moments
Category:Chi-Squared Distribution
Raw Moment of Erlang Distribution

== Theorem ==
Let $n, k$ be strictly positive integers. 
Let $\lambda$ be a strictly positive real number. 
Let $X$ have a continuous random variable with an Erlang distribution with parameters $k$ and $\lambda$.
Then the $n$th raw moment of $X$ is given by:
:$\ds \expect {X^n} = \frac 1 {\lambda^n} \prod_{m \mathop = 0}^{n - 1} \paren {k + m}$
 
== Proof ==
From the definition of the Erlang distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac {\lambda^k x^{k - 1} e^{- \lambda x} } {\map \Gamma k}$
From the definition of the expected value of a continuous random variable: 
:$\ds \expect {X^n} = \int_0^\infty x^n \map {f_X} x \rd x$
So:
Category:Raw Moments
Category:Erlang Distribution
Raw Moment of Exponential Distribution

== Theorem ==
Let $X$ be a continuous random variable of the exponential distribution with parameter $\beta$ for some $\beta \in \R_{>0}$.
Let $n$ be a strictly positive integer. 
Then the $n$th raw moment of $X$ is given by: 
:$\expect {X^n} = n! \beta^n$
== Proof == 
From Moment Generating Function of Exponential Distribution, the moment generating function of $X$ is given by:
:$\map {M_X} t = \dfrac 1 {1 - \beta t}$
By Moment in terms of Moment Generating Function:
:$\expect {X^n} = \map {M^{\paren n}_X} 0$
We have: 
Setting $t = 0$ gives:
Category:Exponential Distribution
Category:Raw Moments
Raw Moment of Log Normal Distribution

== Theorem ==
Let $X$ be a continuous random variable with the Log Normal distribution with $\mu \in \R, \sigma \in \R_{>0}$.
Let $n$ be a strictly positive integer. 
Then the $n$th raw moment $\expect {X^n}$ of $X$ is given by: 
:$\expect {X^n} = \exp {\paren {n\mu + \dfrac {\sigma^2 n^2} 2} }$
== Proof == 
From the definition of the Log Normal distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac 1 {\sigma \sqrt {2 \pi} x} \map \exp {-\dfrac {\paren {\map \ln x - \mu}^2} {2 \sigma^2} }$
where $\Img X = \R_{>0}$.
From the definition of the expected value of a continuous random variable: 
:$\ds \expect {X^n} = \int_0^\infty x^n \map {f_X} x \rd x$
Therefore:
Let: 
:$u = \dfrac {\paren {\map \ln x - \mu}} {\sqrt 2 \sigma}$
Then by Chain Rule for Derivatives, we have:
:$\dfrac {\d u} {\d x} = \dfrac 1 {x \sqrt 2 \sigma}$
and also:
:$\exp {\paren {\sqrt 2 \sigma nu + n \mu} } = x^n$
We can see that:
:$u \to -\infty$ as $x \to 0$
and
:$u \to \infty$ as $x \to \infty$
Plugging these results back into our integral above, we have:
Category:Log Normal Distribution
Category:Raw Moments
Raw Moment of Pareto Distribution

== Theorem ==
Let $X$ be a continuous random variable with the Pareto distribution with $a, b \in \R_{> 0}$.
Let $n$ be a strictly positive integer. 
Then the $n$th raw moment $\expect {X^n}$ of $X$ is given by: 
:$\expect {X^n} = \begin {cases} \dfrac {a b^n} {a - n} & n < a \\ \text {does not exist} & n \ge a \end {cases}$
== Proof == 
From the definition of the Pareto distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac {a b^a} {x^{a + 1} }$
Where $\Img X \in \hointr b \infty$.
From the definition of the expected value of a continuous random variable: 
:$\ds \expect {X^n} = \int_b^\infty x^n \map {f_X} x \rd x$
First take $a > n$. 
Now take $a = n$.
Finally, take $a < n$. 
Category:Pareto Distribution
Category:Raw Moments
Raw Moment of Poisson Distribution

== Theorem ==
Let $X$ be a discrete random variable with the Poisson distribution with parameter $\lambda$.
Let $n$ be a strictly positive integer. 
Then the $n$th raw moment $\expect {X^n}$ of $X$ is given by: 
:$\ds \expect {X^n} = \sum_{k \mathop = 0}^n \lambda^k {n \brace k}$
where $\ds {n \brace k}$ is a Stirling number of the second kind.
== Proof == 
Category:Poisson Distribution
Category:Raw Moments