Proof: The product of independent log-normal random variables is a log-normal random variable
Theorem: Let $X_1, \ldots, X_n$ be independent random variables following log-normal distributions:
\[\label{eq:X-lognorm} X_i \sim \ln \mathcal{N}(\mu_i, \sigma_i^2), \; i = 1, \ldots, n \; .\]Then, the product of these random variables also follows a log-normal distribution
\[\label{eq:Z-lognorm} Z = \prod_{i=1}^n X_i \sim \mathcal{N}(\mu, \sigma^2)\]where the log-normal 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:
\[\label{eq:lognorm-norm} X \sim \ln \mathcal{N}(\mu, \sigma^2) \quad \Leftrightarrow \quad \ln X \sim \mathcal{N}(\mu, \sigma^2) \; .\]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 random variables is a sum of independent normal random variables. This sum, like any linear combination of independent normal random variables, is again normally distributed. 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}.
- drhab (2018): "If Xi is a log-normal r.v., show product of all Xi is also a log-normal r.v."; in: StackExchange Mathematics, retrieved on 2024-11-15; URL: https://math.stackexchange.com/a/2838307/480910.
Metadata: ID: P479 | shortcut: lognorm-prodind | author: JoramSoch | date: 2024-11-15, 11:52.