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Let $X$ be a normal random variable with mean $\mu$ and variance $\sigma^2,\; X\sim N(\mu, \sigma^2).$ Prove that the variable $Y = \exp(X)$ has a log-normal distribution. $$f(y)=\frac{1}{y\sigma\sqrt{2\pi}}\,\exp\left(-\frac{(\ln(y)-\mu)^2}{2\sigma^2}\right)$$

I've tried to plug the inverse function of $Y(x = \ln(y))$ into the pdf of $X$ (normal distribution), but it did not work.

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You don’t just use the transformation in the PDF. Start with the CDF $F_Y(y)=P(Y\le y)=P(e^X\le y)=P(X\le \log(y))$ and see what you can do that that. Since this is a self-study question, it would be best for you to work through the details before you see someone else’s derivation.

In some sense, taking derivatives (don’t forget the chain rule) of CDFs is more natural than integrating PDFs.

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