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Merge pull request #92 from JoramSoch/master
added 3 definitions and 3 proofs
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‎D/mom-cent.md‎

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---
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layout: definition
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mathjax: true
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author: "Joram Soch"
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affiliation: "BCCN Berlin"
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e_mail: "joram.soch@bccn-berlin.de"
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date: 2020-10-08 03:37:00
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title: "Central moment"
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chapter: "General Theorems"
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section: "Probability theory"
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topic: "Further moments"
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definition: "Central moment"
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sources:
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- authors: "Wikipedia"
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year: 2020
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title: "Moment (mathematics)"
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in: "Wikipedia, the free encyclopedia"
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pages: "retrieved on 2020-10-08"
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url: "https://en.wikipedia.org/wiki/Moment_(mathematics)#Significance_of_the_moments"
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def_id: "D98"
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shortcut: "mom-cent"
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username: "JoramSoch"
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---
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**Definition:** Let $X$ be a [random variable](/D/rvar) with [expected value](/D/mean) $\mu$ and let $n$ be a positive integer. Then, the $n$-th central moment of $X$ is defined as the $n$-th [moment](/D/mom) of $X$ about the value $\mu$:
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$$ \label{eq:mom-cent}
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\mu_n = \mathrm{E}[(X-\mu)^n] \; .
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$$

‎D/mom-raw.md‎

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---
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layout: definition
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mathjax: true
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author: "Joram Soch"
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affiliation: "BCCN Berlin"
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e_mail: "joram.soch@bccn-berlin.de"
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date: 2020-10-08 03:31:00
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title: "Raw moment"
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chapter: "General Theorems"
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section: "Probability theory"
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topic: "Further moments"
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definition: "Raw moment"
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sources:
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- authors: "Wikipedia"
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year: 2020
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title: "Moment (mathematics)"
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in: "Wikipedia, the free encyclopedia"
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pages: "retrieved on 2020-10-08"
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url: "https://en.wikipedia.org/wiki/Moment_(mathematics)#Significance_of_the_moments"
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def_id: "D97"
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shortcut: "mom-raw"
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username: "JoramSoch"
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---
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**Definition:** Let $X$ be a [random variable](/D/rvar) and let $n$ be a positive integer. Then, the $n$-th raw moment of $X$, also called ($n$-th) "crude moment", is defined as the $n$-th [moment](/D/mom) of $X$ about the value 0:
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$$ \label{eq:mom-raw}
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\mu_n' = \mu_n(0) = \mathrm{E}[(X-0)^n] = \mathrm{E}[X^n] \; .
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$$

‎D/mom-stand.md‎

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---
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layout: definition
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mathjax: true
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author: "Joram Soch"
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affiliation: "BCCN Berlin"
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e_mail: "joram.soch@bccn-berlin.de"
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date: 2020-10-08 03:47:00
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title: "Standardized moment"
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chapter: "General Theorems"
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section: "Probability theory"
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topic: "Further moments"
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definition: "Standardized moment"
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sources:
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- authors: "Wikipedia"
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year: 2020
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title: "Moment (mathematics)"
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in: "Wikipedia, the free encyclopedia"
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pages: "retrieved on 2020-10-08"
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url: "https://en.wikipedia.org/wiki/Moment_(mathematics)#Standardized_moments"
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def_id: "D99"
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shortcut: "mom-stand"
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username: "JoramSoch"
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---
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**Definition:** Let $X$ be a [random variable](/D/rvar) with [expected value](/D/mean) $\mu$ and [standard deviation](/D/std) $\sigma$ and let $n$ be a positive integer. Then, the $n$-th standardized moment of $X$ is defined as the $n$-th [moment](/D/mom) of $X$ about the value $\mu$, divided by the $n$-th power of $\sigma$:
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$$ \label{eq:mom-stand}
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\mu_n^{*} = \frac{\mu_n}{\sigma^n} = \frac{\mathrm{E}[(X-\mu)^n]}{\sigma^n} \; .
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$$

‎I/Table_of_Contents.md‎

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1.10. Further moments <br>
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&emsp;&ensp; 1.10.1. *[Moment](/D/mom)* <br>
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&emsp;&ensp; 1.10.2. **[Moment in terms of moment-generating function](/P/mom-mgf)** <br>
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&emsp;&ensp; 1.10.3. **[First central moment is zero](/P/momcent-1st)** <br>
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&emsp;&ensp; 1.10.3. *[Raw moment](/D/mom-raw)* <br>
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&emsp;&ensp; 1.10.4. **[First raw moment is mean](/P/momraw-1st)** <br>
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&emsp;&ensp; 1.10.5. **[Second raw moment and variance](/P/momraw-2nd)** <br>
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&emsp;&ensp; 1.10.6. *[Central moment](/D/mom-cent)* <br>
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&emsp;&ensp; 1.10.7. **[First central moment is zero](/P/momcent-1st)** <br>
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&emsp;&ensp; 1.10.8. **[Second central moment is variance](/P/momcent-2nd)** <br>
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&emsp;&ensp; 1.10.9. *[Standardized moment](/D/mom-stand)* <br>
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2. Information theory
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‎P/momcent-2nd.md‎

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---
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layout: proof
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mathjax: true
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author: "Joram Soch"
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affiliation: "BCCN Berlin"
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e_mail: "joram.soch@bccn-berlin.de"
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date: 2020-10-08 05:13:00
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title: "Second central moment is variance"
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chapter: "General Theorems"
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section: "Probability theory"
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topic: "Further moments"
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theorem: "Second central moment is variance"
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sources:
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- authors: "Wikipedia"
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year: 2020
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title: "Moment (mathematics)"
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in: "Wikipedia, the free encyclopedia"
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pages: "retrieved on 2020-10-08"
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url: "https://en.wikipedia.org/wiki/Moment_(mathematics)#Significance_of_the_moments"
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proof_id: "P173"
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shortcut: "momcent-2nd"
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username: "JoramSoch"
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---
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**Theorem:** The second [central moment](/D/mom-cent) equals the [variance](/D/var), i.e.
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$$ \label{eq:momcent-2nd}
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\mu_2 = \mathrm{Var}(X) \; .
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$$
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**Proof:** The second [central moment](/D/mom-cent) of a [random variable](/D/rvar) $X$ with [mean](/D/mean) $\mu$ is defined as
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$$ \label{eq:momcent-2nd-def}
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\mu_2 = \mathrm{E}\left[ (X-\mu)^2 \right]
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$$
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which is equivalent to the [definition of the variance](/D/var):
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$$ \label{eq:momraw-1st-qed}
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\mu_2 = \mathrm{E}\left[ (X - \mathrm{E}(X))^2 \right] = \mathrm{Var}(X) \; .
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$$

‎P/momraw-1st.md‎

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---
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layout: proof
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mathjax: true
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author: "Joram Soch"
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affiliation: "BCCN Berlin"
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e_mail: "joram.soch@bccn-berlin.de"
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date: 2020-10-08 04:19:00
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title: "First raw moment is mean"
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chapter: "General Theorems"
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section: "Probability theory"
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topic: "Further moments"
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theorem: "First raw moment is mean"
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sources:
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proof_id: "P171"
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shortcut: "momraw-1st"
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username: "JoramSoch"
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---
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**Theorem:** The first [raw moment](/D/mom-raw) equals the [mean](/D/mean), i.e.
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$$ \label{eq:momraw-1st}
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\mu_1' = \mu \; .
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$$
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**Proof:** The first [raw moment](/D/mom-raw) of a [random variable](/D/rvar) $X$ is defined as
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$$ \label{eq:momraw-1st-def}
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\mu_1' = \mathrm{E}\left[ (X-0)^1 \right]
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$$
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which is equal to the [expected value](/D/mean) of $X$:
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$$ \label{eq:momraw-1st-qed}
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\mu_1' = \mathrm{E}\left[ X \right] = \mu \; .
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$$

‎P/momraw-2nd.md‎

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---
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layout: proof
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mathjax: true
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author: "Joram Soch"
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affiliation: "BCCN Berlin"
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e_mail: "joram.soch@bccn-berlin.de"
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date: 2020-10-08 05:05:00
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title: "Relationship between second raw moment, variance and mean"
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chapter: "General Theorems"
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section: "Probability theory"
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topic: "Further moments"
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theorem: "Second raw moment and variance"
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sources:
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proof_id: "P172"
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shortcut: "momraw-2nd"
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username: "JoramSoch"
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---
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**Theorem:** The second [raw moment](/D/mom-raw) can be expressed as
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$$ \label{eq:momraw-2nd}
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\mu_2' = \mathrm{Var}(X) + \mathrm{E}(X)^2
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$$
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where $\mathrm{Var}(X)$ is the [variance](/D/var) of $X$ and $\mathrm{E}(X)$ is the [expected value](/D/mean) of $X$.
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**Proof:** The second [raw moment](/D/mom-raw) of a [random variable](/D/rvar) $X$ is defined as
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$$ \label{eq:momraw-2nd-def}
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\mu_2' = \mathrm{E}\left[ (X-0)^2 \right] \; .
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$$
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Using the [partition of variance into expected values](/P/var-mean)
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$$ \label{eq:var-mean}
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\mathrm{Var}(X) = \mathrm{E}(X^2) - \mathrm{E}(X)^2 \; ,
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$$
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the second raw moment can be rearranged into:
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$$ \label{eq:momraw-2nd-qed}
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\mu_2' \overset{\eqref{eq:momraw-2nd-def}}{=} \mathrm{E}(X^2) \overset{\eqref{eq:var-mean}}{=} \mathrm{Var}(X) + \mathrm{E}(X)^2 \; .
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$$

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