Short Derivation Of Log Normal Distribution

If \(X~LN(\mu,\sigma^2)\) then \(ln(X)=Y\) is \(Y\) is distributed \(N(\mu,\sigma^2)\).

Derivation of log-normal distribution

\(\begin{align}
Pr(X < k) &= Pr(e^{Y} < k) \\
&= Pr(Y < ln(k)) \\
&= \int_{\infty}^{ln(k)} \frac{1}{\sqrt{2\pi \sigma}} e^{- \frac{(Y-\mu)^2}{2\sigma^2}} dy \\
&= \int_{\infty}^{ln(k)} \frac{1}{\sqrt{2\pi \sigma}} e^{- \frac{(ln(x)-\mu)^2}{2\sigma^2}} \frac{1}{x} \frac{dx}{dy}dy \\
&= \int_{\infty}^{ln(k)} \frac{1}{x\sqrt{2\pi \sigma}} e^{- \frac{(ln(x)-\mu)^2}{2\sigma^2}} dx \end{align}\)