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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.

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  • $\begingroup$ It might be worth starting by finding the mean and variance of $X$ $\endgroup$ – Henry Jan 31 at 0:31
  • $\begingroup$ $E(X)=\sqrt{e}$, $E(X^2)=e^2$, and $Var(X)=e(e-1)$ $\endgroup$ – ddswsd Jan 31 at 2:04
  • $\begingroup$ 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 $\endgroup$ – Henry Jan 31 at 8:11

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