Log-normal Distribution Inverse CDF Calculator

Log-normal Distribution Inverse CDF Calculator

For a log‑normal distribution with parameters \(\mu\) and \(\sigma\), the inverse CDF (quantile function) is: $$ Q(p; \mu, \sigma) = \exp\Bigl(\mu + \sigma\,\Phi^{-1}(p)\Bigr), $$ where \(\Phi^{-1}(p)\) is the inverse standard normal CDF.

* Enter a probability \( p \) (with \( 0 < p < 1 \)), the location parameter \(\mu\), and the scale parameter \(\sigma > 0\).

Step 1: Enter Parameters

Enter a probability between 0 and 1 (e.g., 0.5).

e.g., 0

e.g., 1 (must be > 0)

How It Works

The log‑normal distribution is defined as the distribution of a random variable whose logarithm is normally distributed.

If \(\ln(X) \sim N(\mu, \sigma^2)\), then the CDF of \( X \) is:

$$ F(x;\mu,\sigma) = \Phi\!\Bigl(\frac{\ln(x)-\mu}{\sigma}\Bigr), $$

To compute the inverse CDF (quantile function), we solve for \( x \) such that: $$ p = \Phi\!\Bigl(\frac{\ln(x)-\mu}{\sigma}\Bigr). $$ Taking the inverse of \(\Phi\) gives: $$ x = \exp\Bigl(\mu + \sigma\,\Phi^{-1}(p)\Bigr). $$

Formula: \( Q(p; \mu, \sigma) = \exp\Bigl(\mu + \sigma\,\Phi^{-1}(p)\Bigr) \)

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