Merge pull request #273 from JoramSoch/master

added 1 proof and 1 definition

Author: JoramSoch

D/para.md

@@ -0,0 +1,32 @@
+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2024-09-27 12:40:05
+
+title: "Parameter"
+chapter: "General Theorems"
+section: "Probability theory"
+topic: "Probability distributions"
+definition: "Statistical parameter"
+
+sources:
+  - authors: "Wikipedia"
+    year: 2024
+    title: "Statistical parameter"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2024-09-27"
+    url: "https://en.wikipedia.org/wiki/Statistical_parameter#Parameterised_distributions"
+
+def_id: "D204"
+shortcut: "para"
+username: "JoramSoch"
+---
+
+
+**Definition:** A parameter, also "statistical parameter", is any fixed quantity, i.e. [constant](/D/const) scalar, vector or matrix, that describes a parametrized [probability distribution](/D/dist).
+
+Examples of parameters include the mean and variance parameters of a [normal distribution](/D/norm), covariance parameters in a [multivariate](/D/mvn) or [matrix](/D/matn)-normal distribution, shape and rate parameters of the [gamma distribution](/D/gam) or the vector of category probabilities in a [multinomial distribution](/D/mult).

I/ToC.md

@@ -68,6 +68,7 @@ title: "Table of Contents"
    &emsp;&ensp; 1.5.3. *[Marginal distribution](/D/dist-marg)* <br>
    &emsp;&ensp; 1.5.4. *[Conditional distribution](/D/dist-cond)* <br>
    &emsp;&ensp; 1.5.5. *[Sampling distribution](/D/dist-samp)* <br>
+   &emsp;&ensp; 1.5.6. *[Statistical parameter](/D/para)* <br>
 
    <p id="Probability mass function"></p>
    1.6. Probability mass function <br>
@@ -184,10 +185,11 @@ title: "Table of Contents"
    1.14. Correlation <br>
    &emsp;&ensp; 1.14.1. *[Definition](/D/corr)* <br>
    &emsp;&ensp; 1.14.2. **[Range](/P/corr-range)** <br>
-   &emsp;&ensp; 1.14.3. *[Sample correlation coefficient](/D/corr-samp)* <br>
-   &emsp;&ensp; 1.14.4. **[Relationship to standard scores](/P/corr-z)** <br>
-   &emsp;&ensp; 1.14.5. *[Correlation matrix](/D/corrmat)* <br>
-   &emsp;&ensp; 1.14.6. *[Sample correlation matrix](/D/corrmat-samp)* <br>
+   &emsp;&ensp; 1.14.3. **[Correlation under independence](/P/corr-ind)** <br>
+   &emsp;&ensp; 1.14.4. *[Sample correlation coefficient](/D/corr-samp)* <br>
+   &emsp;&ensp; 1.14.5. **[Relationship to standard scores](/P/corr-z)** <br>
+   &emsp;&ensp; 1.14.6. *[Correlation matrix](/D/corrmat)* <br>
+   &emsp;&ensp; 1.14.7. *[Sample correlation matrix](/D/corrmat-samp)* <br>
 
    <p id="Measures of central tendency"></p>
    1.15. Measures of central tendency <br>

P/corr-ind.md

@@ -0,0 +1,49 @@
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2024-09-27 12:32:50
+
+title: "Independent random variables are uncorrelated"
+chapter: "General Theorems"
+section: "Probability theory"
+topic: "Correlation"
+theorem: "Correlation under independence"
+
+sources:
+  - authors: "StatProofBook"
+    year: 2022
+    title: "Uncorrelated random variables are not necessarily independent."
+    in: "X"
+    pages: "Nov 22, 2022, 06:34 AM"
+    url: "https://x.com/StatProofBook/status/1594927275514134528"
+
+proof_id: "P472"
+shortcut: "corr-ind"
+username: "JoramSoch"
+---
+
+
+**Theorem:** [Independent](/D/ind) [random variables](/D/rvar) are uncorrelated.
+
+
+**Proof:** The [correlation](/D/corr) of two random variables is defined as:
+
+$$ \label{eq:corr}
+\mathrm{Corr}(X,Y) = \frac{\mathrm{Cov}(X,Y)}{\sqrt{\mathrm{Var}(X)} \sqrt{\mathrm{Var}(Y)}} \; .
+$$
+
+The [covariance of independent random variables is zero](/P/cov-ind):
+
+$$ \label{eq:cov-ind}
+X, Y \; \text{independent} \quad \Rightarrow \quad \mathrm{Cov}(X,Y) = 0 \; .
+$$
+
+Thus, the correlation of independent random variables is also zero:
+
+$$ \label{eq:corr-ind-qed}
+X, Y \; \text{independent} \quad \Rightarrow \quad \mathrm{Corr}(X,Y) = 0 \; .
+$$