I was thinking about the meaning of location-scale family. My understanding is that for every $X$ member of a location scale family with parameters $a$ location and $b$ scale, then the distribution of $Z =(X-a)/b$ does not depend of any parameters and it's the same for every $X$ belonging to that family.
So my question is could you provide an example where two random from the same distribution family are standardized but that does not results in a Random Variable with the same distribution?
Say $X$ and $Y$ come from the same distribution family (where with family I mean for example both Normal or both Gamma and so on ..). Define:
$Z_1 = \dfrac{X-\mu}{\sigma}$
$Z_2 = \dfrac{Y-\mu}{\sigma}$
we know that both $Z_1$ and $Z_2$ have the same expectation and variance, $\mu_Z =0, \sigma^2_Z =1$.
But can they have different higher moments?
My attempt to answer this question is that if the distribution of $X$ and $Y$ depends on more than 2 parameters than it could be. And I am thinking about the generalized $t-student$ that has 3 parameters.
But if the number of parameters is $\le2$ and $X$ and $Y$ come from the same distribution family with the same expectation and variance, then does it mean that $Z_1$ and $Z_2$ has the same distribution (higher moments)?
Thank you