added 2 definitions

Author: JoramSoch

D/corr-samp.md

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+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2021-12-14 07:23:00
+
+title: "Sample correlation coefficient"
+chapter: "General Theorems"
+section: "Probability theory"
+topic: "Correlation"
+definition: "Sample correlation coefficient"
+
+sources:
+  - authors: "Wikipedia"
+    year: 2021
+    title: "Pearson 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"
+
+def_id: "D168"
+shortcut: "corr-samp"
+username: "JoramSoch"
+---
+
+
+**Definition:** 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 of $x$ and $y$ is given by
+
+$$ \label{eq:corr-samp}
+r_{xy} = \frac{\sum_{i=1}^n (x_i-\bar{x}) (y_i-\bar{y})}{\sqrt{\sum_{i=1}^n (x_i-\bar{x})^2} \sqrt{\sum_{i=1}^n (y_i-\bar{y})^2}}
+$$
+
+where $\bar{x}$ and $\bar{y}$ are the [sample means](/D/mean-samp).

D/corrmat-samp.md

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+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2021-12-14 07:45:00
+
+title: "Sample correlation matrix"
+chapter: "General Theorems"
+section: "Probability theory"
+topic: "Correlation"
+definition: "Sample correlation matrix"
+
+sources:
+
+def_id: "D169"
+shortcut: "corrmat-samp"
+username: "JoramSoch"
+---
+
+
+**Definition:** Let $x = \left\lbrace x_1, \ldots, x_n \right\rbrace$ be a [sample](/D/samp) from a [random vector](/D/rvec) $X \in \mathbb{R}^{p \times 1}$. Then, the sample correlation matrix of $x$ is the matrix whose entries are the [sample correlation coefficients](/D/corr-samp) between pairs of entries of $x_1, \ldots, x_n$:
+
+$$ \label{eq:corrmat-samp-v1}
+\mathrm{R}_{xx} =
+\begin{bmatrix}
+r_{x^{(1)},x^{(1)}} & \ldots & r_{x^{(1)},x^{(n)}} \\
+\vdots & \ddots & \vdots \\
+r_{x^{(n)},x^{(1)}} & \ldots & r_{x^{(n)},x^{(n)}}
+\end{bmatrix}
+$$
+
+where the $r_{x^{(j)},x^{(k)}}$ is the [sample correlation](/D/corr-samp) between the $j$-th and the $k$-th entry of $X$ given by
+
+$$ \label{eq:corrmat-samp-v2}
+r_{x^{(j)},x^{(k)}} = \frac{\sum_{i=1}^n (x_{ij}-\bar{x}^{(j)}) (x_{ik}-\bar{x}^{(k)})}{\sqrt{\sum_{i=1}^n (x_{ij}-\bar{x}^{(j)})^2} \sqrt{\sum_{i=1}^n (x_{ik}-\bar{x}^{(k)})^2}}
+$$
+
+in which $\bar{x}^{(j)}$ and $\bar{x}^{(k)}$ are the [sample means](/D/mean-samp)
+
+$$ \label{eq:mean-samp}
+\begin{split}
+\bar{x}^{(j)} &= \frac{1}{n} \sum_{i=1}^n x_{ij} \\
+\bar{x}^{(k)} &= \frac{1}{n} \sum_{i=1}^n x_{ik} \; .
+\end{split}
+$$