added 2 proofs

Author: JoramSoch

P/corr-range.md

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+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2021-12-14 02:08:00
+
+title: "Correlation always falls between -1 and +1"
+chapter: "General Theorems"
+section: "Probability theory"
+topic: "Correlation"
+theorem: "Range"
+
+sources:
+  - authors: "Dor Leventer"
+    year: 2021
+    title: "How can I simply prove that the pearson correlation coefficient is between -1 and 1?"
+    in: "StackExchange Mathematics"
+    pages: "retrieved on 2021-12-14"
+    url: "https://math.stackexchange.com/a/4260655/480910"
+
+proof_id: "P300"
+shortcut: "corr-range"
+username: "JoramSoch"
+---
+
+
+**Theorem:** Let $X$ and $Y$ be two [random variables](/D/rvar). Then, the correlation of $X$ and $Y$ is between and including $-1$ and $+1$:
+
+$$ \label{eq:corr-range}
+-1 \leq \mathrm{Corr}(X,Y) \leq +1 \; .
+$$
+
+
+**Proof:** Consider the [variance](/D/var) of $X$ plus or minus $Y$, divided by their [standard deviations](/D/std):
+
+$$ \label{eq:var-XY}
+\mathrm{Var}\left( \frac{X}{\sigma_X} \pm \frac{Y}{\sigma_Y} \right) \; .
+$$
+
+Because the [variance is non-negative](/P/var-nonneg), this term is larger than or equal to zero:
+
+$$ \label{eq:var-XY-0}
+0 \leq \mathrm{Var}\left( \frac{X}{\sigma_X} \pm \frac{Y}{\sigma_Y} \right) \; .
+$$
+
+Using the [variance of a linear combination](/P/var-lincomb), it can also be written as:
+
+$$ \label{eq:var-XY-s1}
+\begin{split}
+\mathrm{Var}\left( \frac{X}{\sigma_X} \pm \frac{Y}{\sigma_Y} \right) &= \mathrm{Var}\left( \frac{X}{\sigma_X} \right) + \mathrm{Var}\left( \frac{Y}{\sigma_Y} \right) \pm 2 \, \mathrm{Cov}\left( \frac{X}{\sigma_X}, \frac{Y}{\sigma_Y} \right) \\
+&= \frac{1}{\sigma_X^2} \mathrm{Var}(X) + \frac{1}{\sigma_Y^2} \mathrm{Var}(Y) \pm 2 \, \frac{1}{\sigma_X \sigma_Y} \, \mathrm{Cov}(X,Y) \\
+&= \frac{1}{\sigma_X^2} \sigma_X^2 + \frac{1}{\sigma_Y^2} \sigma_Y^2 \pm 2 \, \frac{1}{\sigma_X \sigma_Y} \, \sigma_{XY} \; .
+\end{split}
+$$
+
+Using the [relationship between covariance and correlation](/P/cov-corr), we have:
+
+$$ \label{eq:var-XY-s2}
+\mathrm{Var}\left( \frac{X}{\sigma_X} \pm \frac{Y}{\sigma_Y} \right) = 1 + 1 + \pm 2 \, \mathrm{Corr}(X,Y) \; .
+$$
+
+Thus, the combination of \eqref{eq:var-XY-0} with \eqref{eq:var-XY-s2} yields
+
+$$ \label{eq:var-XY-ineq}
+0 \leq 2 \pm 2 \, \mathrm{Corr}(X,Y)
+$$
+
+which is equivalent to
+
+$$ \label{eq:corr-range-qed}
+-1 \leq \mathrm{Corr}(X,Y) \leq +1 \; .
+$$

P/corr-z.md

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+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2021-12-14 02:31:00
+
+title: "Correlation coefficient in terms of standard scores"
+chapter: "General Theorems"
+section: "Probability theory"
+topic: "Correlation"
+theorem: "Relationship to standard scores"
+
+sources:
+  - authors: "Wikipedia"
+    year: 2021
+    title: "Peason correlation coefficient"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2021-12-14"
+    url: "https://en.wikipedia.org/wiki/Pearson_correlation_coefficient#For_a_sample"
+
+proof_id: "P299"
+shortcut: "corr-z"
+username: "JoramSoch"
+---
+
+
+**Theorem:** Let $x = \left\lbrace x_1, \ldots, x_n \right\rbrace$ and $y = \left\lbrace y_1, \ldots, y_n \right\rbrace$ be [samples](/D/samp) from [random variables](/D/rvar) $X$ and $Y$. Then, the [sample correlation coefficient](/D/corr-samp) $r_{xy}$ can be expressed in terms of the [standard scores](/D/z) of $x$ and $y$:
+
+$$ \label{eq:corr-z}
+r_{xy} = \frac{1}{n-1} \sum_{i=1}^n z_i^{(x)} \cdot z_i^{(y)} = \frac{1}{n-1} \sum_{i=1}^n \left( \frac{x_i-\bar{x}}{s_x} \right) \left( \frac{y_i-\bar{y}}{s_y} \right)
+$$
+
+where $\bar{x}$ and $\bar{y}$ are the [sample means](/D/mean-samp) and $s_x$ and $s_y$ are the [sample variances](/D/var-samp).
+
+
+**Proof:** The [sample correlation coefficient](/D/corr-samp) is defined as
+
+$$ \label{eq:corr-samp}
+r_{xy} = \frac{\sum_{i=1}^n (x_i-\bar{x}) (y_i-\bar{y})}{\sqrt{\sum_{i=1}^n (x_i-\bar{x})^2} \sqrt{\sum_{i=1}^n (y_i-\bar{y})^2}} \; .
+$$
+
+Using the [sample variances](/D/var-samp) of $x$ and $y$, we can write:
+
+$$ \label{eq:corr-z-s1}
+r_{xy} = \frac{\sum_{i=1}^n (x_i-\bar{x}) (y_i-\bar{y})}{\sqrt{(n-1) s_x^2} \sqrt{(n-1) s_y^2}} \; .
+$$
+
+Rearranging the terms, we arrive at:
+
+$$ \label{eq:corr-z-s2}
+r_{xy} = \frac{1}{(n-1) \, s_x \, s_y} \sum_{i=1}^n (x_i-\bar{x}) (y_i-\bar{y}) \; .
+$$
+
+Further simplifying, the result is:
+
+$$ \label{eq:corr-z-s3}
+r_{xy} = \frac{1}{n-1} \sum_{i=1}^n \left( \frac{x_i-\bar{x}}{s_x} \right) \left( \frac{y_i-\bar{y}}{s_y} \right) \; .
+$$