Let X∼N(μ,σ2)X \sim N(\mu, \sigma^2)X∼N(μ,σ2). Which is the PDF of Y=eXY = e^XY=eX (lognormal)?
fY(y)=1yσ2πe−(lny−μ)2/(2σ2)f_Y(y) = \frac{1}{y\sigma\sqrt{2\pi}} e^{-(\ln y - \mu)^2/(2\sigma^2)}fY(y)=yσ2π1e−(lny−μ)2/(2σ2)
fY(y)=1σ2πey−eyf_Y(y) = \frac{1}{\sigma\sqrt{2\pi}} e^{y - e^y}fY(y)=σ2π1ey−ey
fY(y)=eyf_Y(y) = e^yfY(y)=ey
fY(y)=12πe−y2/2f_Y(y) = \frac{1}{\sqrt{2\pi}} e^{-y^2/2}fY(y)=2π1e−y2/2