I am really struggling with the definition of a location-scale family. I have an intuition about it, but I cannot answer the questions about it. I want to show that the exponential distribution belongs to the location-scale family associated with the standard exponential distribution $x\mapsto 1-e^{-x}$. Can someone help me with this?
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Exponential distribution does not belong to a location - scale family. Actually it belongs to a "scale family"
Definition
A scale family of distributions has densities of the form
$$ \bbox[5px,border:2px solid black] { g(x|\theta)=\frac{1}{\theta}\cdot\psi\left(\frac{x}{\theta} \right) \qquad (1) } $$
where $\theta>0$ and $\psi$ is a density.
Now let's take the pdf of an exponential rv.
$$f_X(x)=\frac{1}{\theta}e^{-x/\theta}$$
It is immediate to recognize that $f_X(x)$ is in the form (1) where $\psi=e^{-y}$ is the density of a negative exponential rv with mean 1
Note that $\theta$ in this density is called "Scale parameter" exactly for this reason.
as a further example,
$X\sim N(\mu;\sigma^2)$ belongs to a location - scale family.
$X\sim N(\mu;\mu^2)$ belongs to a scale family.
$X\sim N(1;\theta)$ does not belong to a scale family.
Try to prove it using the corresponding definitions
