Merge pull request #280 from JoramSoch/master

added 1 definition and 1 proof

Author: JoramSoch

D/samp.md

@@ -0,0 +1,36 @@
+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2024-11-15 11:46:35
+
+title: "Sample"
+chapter: "General Theorems"
+section: "Probability theory"
+topic: "Random experiments"
+definition: "Statistical sample"
+
+sources:
+  - authors: "Elliott, Graham"
+    year: 2024
+    title: "Sampling"
+    in: "Understanding Statistics"
+    pages: "retrieved on 2024-11-15"
+    url: "https://econweb.ucsd.edu/~gelliott/Chapter5.html"
+
+def_id: "D210"
+shortcut: "samp"
+username: "JoramSoch"
+---
+
+
+**Definition:** Let $X$ be a [random variable](/D/rvar) or a set of [random variables](/D/rvar). Then, a sample, also "statistical sample", is a set of realizations of this random variable, obtained via [observation and measurement](/D/data) within [random experiments](/D/rexp):
+
+$$ \label{eq:samp}
+x = \left\lbrace x_1, \ldots, x_n \right\rbrace \; .
+$$
+
+The number of elements in a sample is called "[sample size](/D/samp-size)".

I/ToC.md

@@ -21,7 +21,8 @@ title: "Table of Contents"
    &emsp;&ensp; 1.1.3. *[Event space](/D/eve-spc)* <br>
    &emsp;&ensp; 1.1.4. *[Probability space](/D/prob-spc)* <br>
    &emsp;&ensp; 1.1.5. *[Measured data](/D/data)* <br>
-   &emsp;&ensp; 1.1.6. *[Sample statistic](/D/stat)* <br>
+   &emsp;&ensp; 1.1.6. *[Statistical sample](/D/samp)* <br>
+   &emsp;&ensp; 1.1.7. *[Sample statistic](/D/stat)* <br>
 
    <p id="Random variables"></p>
    1.2. Random variables <br>
@@ -545,6 +546,7 @@ title: "Table of Contents"
    &emsp;&ensp; 3.6.6. **[Median](/P/lognorm-med)** <br>
    &emsp;&ensp; 3.6.7. **[Mode](/P/lognorm-mode)** <br>
    &emsp;&ensp; 3.6.8. **[Variance](/P/lognorm-var)** <br>
+   &emsp;&ensp; 3.6.9. **[Product of independent log-normals](/P/lognorm-prodind)** <br>
 
    <p id="Chi-squared distribution"></p>
    3.7. Chi-squared distribution <br>

P/lognorm-prodind.md

@@ -0,0 +1,98 @@
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2024-11-15 11:52:45
+
+title: "The product of independent log-normal random variables is a log-normal random variable"
+chapter: "Probability Distributions"
+section: "Univariate continuous distributions"
+topic: "Log-normal distribution"
+theorem: "Product of independent log-normals"
+
+sources:
+  - authors: "drhab"
+    year: 2018
+    title: "If Xi is a log-normal r.v., show product of all Xi is also a log-normal r.v."
+    in: "StackExchange Mathematics"
+    pages: "retrieved on 2024-11-15"
+    url: "https://math.stackexchange.com/a/2838307/480910"
+
+proof_id: "P479"
+shortcut: "lognorm-prodind"
+username: "JoramSoch"
+---
+
+
+**Theorem:** Let $X_1, \ldots, X_n$ be [independent](/D/ind) [random variables](/D/rvar) following [log-normal distributions](/D/lognorm):
+
+$$ \label{eq:X-lognorm}
+X_i \sim \ln \mathcal{N}(\mu_i, \sigma_i^2), \; i = 1, \ldots, n \; .
+$$
+
+Then, the product of these random variables also follows a [log-normal distribution](/D/lognorm):
+
+$$ \label{eq:Z-lognorm}
+Z = \prod_{i=1}^n X_i \sim \mathcal{N}(\mu, \sigma^2)
+$$
+
+where the [log-normal](/D/lognorm) parameters are given by
+
+$$ \label{eq:Z-lognorm-para}
+\mu      = \sum_{i=1}^n \mu_i
+\quad \text{and} \quad
+\sigma^2 = \sum_{i=1}^n \sigma_i^2 \; .
+$$
+
+
+**Proof:** A random variable [follows a log-normal distribution, if and only if its natural logarithm follows a normal distribution](/D/lognorm):
+
+$$ \label{eq:lognorm-norm}
+X \sim \ln \mathcal{N}(\mu, \sigma^2)
+\quad \Leftrightarrow \quad
+\ln X \sim \mathcal{N}(\mu, \sigma^2) \; .
+$$
+
+Thus, from \eqref{eq:X-lognorm}, we have
+
+$$ \label{eq:Y-norm}
+Y_i = \ln X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)
+$$
+
+and from \eqref{eq:Z-lognorm}, we have
+
+$$ \label{eq:ln-Z}
+  \ln Z
+= \ln \left( \prod_{i=1}^n X_i \right)
+= \sum_{i=1}^n \ln X_i
+= \sum_{i=1}^n Y_i \; .
+$$
+
+This means that the logarithm of the product of independent [log-normal](/D/lognorm) random variables is a sum of independent [normal](/D/norm) random variables. This sum, like any [linear combination of independent normal random variables, is again normally distributed](/P/norm-lincomb). Thus, combining \eqref{eq:ln-Z} and \eqref{eq:Y-norm}, we have:
+
+$$ \label{eq:ln-Z-norm}
+     \ln Z
+=    \sum_{i=1}^n Y_i
+\sim \mathcal{N}\left( \sum_{i=1}^n \mu_i, \sum_{i=1}^n \sigma_i^2 \right) \; .
+$$
+
+If a random variable [follows a normal distribution, then its exponential follows a log-normal distribution with the same parameters]:
+
+$$ \label{eq:norm-lognorm}
+Y \sim \mathcal{N}(\mu, \sigma^2)
+\quad \Leftrightarrow \quad
+\exp(Y) \sim \ln \mathcal{N}(\mu, \sigma^2) \; .
+$$
+
+Thus, from \eqref{eq:ln-Z-norm}, we have
+
+$$ \label{eq:Z-lognorm-qed}
+     Z
+=    \exp(\ln Z) 
+\sim \ln \mathcal{N}\left( \sum_{i=1}^n \mu_i, \sum_{i=1}^n \sigma_i^2 \right)
+$$
+
+which is equivalent to \eqref{eq:Z-lognorm} and \eqref{eq:Z-lognorm-para}.