Log‑normal Distribution Variance Calculator
For a log‑normal distribution with parameters \(\mu\) and \(\sigma\) (where \(\ln(X) \sim N(\mu, \sigma^2)\)), the variance is: $$ \operatorname{Var}(X) = \Bigl(\exp\left(\sigma^2\right)-1\Bigr)\exp\Bigl(2\mu+\sigma^2\Bigr). $$
* Enter the location parameter \(\mu\) and the scale parameter \(\sigma\) (with \(\sigma > 0\)).
Step 1: Enter Parameters
e.g., 0
e.g., 1 (must be > 0)
How It Works
The log‑normal distribution is the distribution of a random variable whose logarithm is normally distributed.
If \(\ln(X) \sim N(\mu, \sigma^2)\), then the mean of \(X\) is: $$ E[X] = \exp\!\Bigl(\mu + \frac{\sigma^2}{2}\Bigr), $$ and the variance is: $$ \operatorname{Var}(X) = \Bigl(\exp\left(\sigma^2\right)-1\Bigr)\exp\Bigl(2\mu+\sigma^2\Bigr). $$
Enter the parameters \(\mu\) and \(\sigma\) to compute the variance.
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