# How to prove a variable has a log-normal distribution knowing that the variable is a function of a normal random variable?

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.

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.