### Log-Normal Distribution$X \sim LogN(\mu, \sigma)$

 $\mu=$ $\sigma=$ $x=$ P(X > x) = P(X < x) =

This applet computes probabilities and percentiles for log-normal random variables: $$X \sim LogN(\mu, \sigma)$$

#### Directions

• Enter $\mu$ and $\sigma$.
• To compute a left-tail probability, select $P(X \lt x)$ from the drop-down box, enter a numeric $x$ value in the blue box and press "Tab" or "Enter" on your keyboard. The probability $P(X \lt x)$ will appear in the pink box. Select $P(X \gt x)$ from the drop-down box for a right-tail probability.
• To determine a percentile, enter the percentile (e.g. use 0.8 for the 80th percentile) in the pink box, select $P(X \lt x)$ from the drop-down box, and press "Tab" or "Enter" on your keyboard. The percentile $x$ will appear in the blue box.

On the graph, the $x$ value appears in blue while the probability is shaded in pink.

#### Details

• Probability density function $$f(x)=\frac{1}{x\sqrt{2\pi \sigma^2}} e^{-\frac{1}{2\sigma^2}(\ln(x)-\mu)^2}$$ where $x > 0$, $-\infty < \mu < \infty$, and $\sigma > 0$
• $E(X)=e^{\mu+\sigma^2/2}$
• $Var(X)=(e^{\sigma^2}-1)e^{2\mu+\sigma^2}$
• $SD(X)=\sqrt{(e^{\sigma^2}-1)e^{2\mu+\sigma^2}}$