For $X$~$N(u,v^2)$ and $Y=e^{x}$, is $cY$ lognormal? (where $c>0$ is a constant).
I have already found the pdf of $Y=e^{x}$, which gives us the lognormal distribution.
However, I don't know how to see if $cY$ is lognormal. Am I supposed to plug in $cY$ in for $y$? What does it mean for something to be lognormal?
Any help is greatly appreciated, thank you!
Note that $$cY = ce^X = e^{\log c} e^X = e^{X + \log c}.$$ Since $X + \log c$ is a normal distribution with mean $\mu + \log c$ and variance $\sigma^2$ if $X$ is normal with mean $\mu$ and variance $\sigma^2$, it immediately follows that $cY$ is lognormal with parameters $\mu + \log c$ and $\sigma$.