added 3 proofs

Author: JoramSoch

P/momcent-2nd.md

@@ -0,0 +1,47 @@
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2020-10-08 05:13:00
+
+title: "Second central moment is variance"
+chapter: "General Theorems"
+section: "Probability theory"
+topic: "Further moments"
+theorem: "Second central moment is variance"
+
+sources:
+  - authors: "Wikipedia"
+    year: 2020
+    title: "Moment (mathematics)"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2020-10-08"
+    url: "https://en.wikipedia.org/wiki/Moment_(mathematics)#Significance_of_the_moments"
+
+proof_id: "P173"
+shortcut: "momcent-2nd"
+username: "JoramSoch"
+---
+
+
+**Theorem:** The second [central moment](/D/mom-cent) equals the [variance](/D/var), i.e.
+
+$$ \label{eq:momcent-2nd}
+\mu_2 = \mathrm{Var}(X) \; .
+$$
+
+
+**Proof:** The second [central moment](/D/mom-cent) of a [random variable](/D/rvar) $X$ with [mean](/D/mean) $\mu$ is defined as
+
+$$ \label{eq:momcent-2nd-def}
+\mu_2 = \mathrm{E}\left[ (X-\mu)^2 \right]
+$$
+
+which is equivalent to the [definition of the variance](/D/var):
+
+$$ \label{eq:momraw-1st-qed}
+\mu_2 = \mathrm{E}\left[ (X - \mathrm{E}(X))^2 \right] = \mathrm{Var}(X) \; .
+$$

P/momraw-1st.md

@@ -0,0 +1,41 @@
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2020-10-08 04:19:00
+
+title: "First raw moment is mean"
+chapter: "General Theorems"
+section: "Probability theory"
+topic: "Further moments"
+theorem: "First raw moment is mean"
+
+sources:
+
+proof_id: "P171"
+shortcut: "momraw-1st"
+username: "JoramSoch"
+---
+
+
+**Theorem:** The first [raw moment](/D/mom-raw) equals the [mean](/D/mean), i.e.
+
+$$ \label{eq:momraw-1st}
+\mu_1' = \mu \; .
+$$
+
+
+**Proof:** The first [raw moment](/D/mom-raw) of a [random variable](/D/rvar) $X$ is defined as
+
+$$ \label{eq:momraw-1st-def}
+\mu_1' = \mathrm{E}\left[ (X-0)^1 \right]
+$$
+
+which is equal to the [expected value](/D/mean) of $X$:
+
+$$ \label{eq:momraw-1st-qed}
+\mu_1' = \mathrm{E}\left[ X \right] = \mu \; .
+$$

P/momraw-2nd.md

@@ -0,0 +1,49 @@
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2020-10-08 05:05:00
+
+title: "Relationship between second raw moment, variance and mean"
+chapter: "General Theorems"
+section: "Probability theory"
+topic: "Further moments"
+theorem: "Second raw moment and variance"
+
+sources:
+
+proof_id: "P172"
+shortcut: "momraw-2nd"
+username: "JoramSoch"
+---
+
+
+**Theorem:** The second [raw moment](/D/mom-raw) can be expressed as
+
+$$ \label{eq:momraw-2nd}
+\mu_2' = \mathrm{Var}(X) + \mathrm{E}(X)^2
+$$
+
+where $\mathrm{Var}(X)$ is the [variance](/D/var) of $X$ and $\mathrm{E}(X)$ is the [expected value](/D/mean) of $X$.
+
+
+**Proof:** The second [raw moment](/D/mom-raw) of a [random variable](/D/rvar) $X$ is defined as
+
+$$ \label{eq:momraw-2nd-def}
+\mu_2' = \mathrm{E}\left[ (X-0)^2 \right] \; .
+$$
+
+Using the [partition of variance into expected values](/P/var-mean)
+
+$$ \label{eq:var-mean}
+\mathrm{Var}(X) = \mathrm{E}(X^2) - \mathrm{E}(X)^2 \; ,
+$$
+
+the second raw moment can be rearranged into:
+
+$$ \label{eq:momraw-2nd-qed}
+\mu_2' \overset{\eqref{eq:momraw-2nd-def}}{=} \mathrm{E}(X^2) \overset{\eqref{eq:var-mean}}{=} \mathrm{Var}(X) + \mathrm{E}(X)^2 \; .
+$$