added proof "lognorm-geomind"

Author: JoramSoch

P/lognorm-geomind.md

@@ -0,0 +1,98 @@
+---
+layout: proof
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2024-11-22 11:39:58
+
+title: "The geometric mean of independent log-normal random variables is a log-normal random variable"
+chapter: "Probability Distributions"
+section: "Univariate continuous distributions"
+topic: "Log-normal distribution"
+theorem: "Geometric mean of independent log-normals "
+
+sources:
+  - authors: "Probability Fact"
+    year: 2022
+    title: "The geometric mean of independent log-normal random variables has a log-normal distribution"
+    in: "X"
+    pages: "retrieved on 2024-11-22"
+    url: "https://x.com/ProbFact/status/1592989704646848512"
+
+proof_id: "P480"
+shortcut: "lognorm-geomind"
+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 [geometric mean](/D/mean-geom) of these random variables also follows a [log-normal distribution](/D/lognorm):
+
+$$ \label{eq:Z-lognorm}
+Z = \sqrt[n]{\prod_{i=1}^n X_i} \sim \mathcal{N}(\mu, \sigma^2)
+$$
+
+where the [log-normal](/D/lognorm) parameters are given by
+
+$$ \label{eq:Z-lognorm-para}
+\mu      = \frac{1}{n} \sum_{i=1}^n \mu_i
+\quad \text{and} \quad
+\sigma^2 = \frac{1}{n^2} \sum_{i=1}^n \sigma_i^2 \; .
+$$
+
+
+**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
+
+$$ \label{eq:ln-Z}
+  \ln Z
+= \ln \left( \sqrt[n]{\prod_{i=1}^n X_i} \right)
+= \frac{1}{n} \sum_{i=1}^n \ln X_i
+= \frac{1}{n} \sum_{i=1}^n Y_i \; .
+$$
+
+This means that the logarithm of the geometric mean of independent [log-normal](/D/lognorm) random variables is the arithmetic mean of independent [normal](/D/norm) random variables. This average, 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:
+
+$$ \label{eq:ln-Z-norm}
+     \ln Z
+=    \frac{1}{n} \sum_{i=1}^n Y_i
+\sim \mathcal{N}\left( \frac{1}{n} \sum_{i=1}^n \mu_i, \, \frac{1}{n^2} \sum_{i=1}^n \sigma_i^2 \right) \; .
+$$
+
+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
+
+$$ \label{eq:Z-lognorm-qed}
+     Z
+=    \exp(\ln Z)
+\sim \ln \mathcal{N}\left( \frac{1}{n} \sum_{i=1}^n \mu_i, \, \frac{1}{n^2} \sum_{i=1}^n \sigma_i^2 \right)
+$$
+
+which is equivalent to \eqref{eq:Z-lognorm} and \eqref{eq:Z-lognorm-para}.