Show how to form a location-scale family $g(x)$ based on $f(x)$ such that $g(x)$ has mean $0$ and variance $1$.

Consider a random variable $$X$$ with the log-normal pdf $$f(x) ={1\over \sqrt{2π}}x^{−1}e^{{−0.5 (logx)^2}}$$, $$x >0$$.

Show how to form a location-scale family $$g(x)$$ based on $$f(x)$$ such that $$g(x)$$ has mean $$0$$ and variance $$1$$.

I'm not sure how to go about this. I know that a location-scale family is a family of probability distributions parametrized by a location parameter and a non-negative scale parameter.

• It might be worth starting by finding the mean and variance of $X$ – Henry Jan 31 at 0:31
• $E(X)=\sqrt{e}$, $E(X^2)=e^2$, and $Var(X)=e(e-1)$ – ddswsd Jan 31 at 2:04
• Then you want $Y=\dfrac{X-\sqrt{e}}{\sqrt{e(e-1)}}$ and you can use the expressions at en.wikipedia.org/wiki/Location%E2%80%93scale_family – Henry Jan 31 at 8:11