I am little confused about deriving the location-scale family of these three distributions.

(a) Let $f(x)=\frac{1}{\pi (1+x^2)}$, $-\infty<x<\infty$. I have no idea how to go about this.

(b) Gamma$(\alpha_0,\beta)$ distributions, where $\alpha_0$ forms a scale family.

My attempt:

The density of a Gamma$(\alpha_0,\beta)$ distribution is $$f(x, \alpha_0) = \frac{1}{\Gamma(\alpha_0)}x^{\alpha_0 - 1}e^{x}, \quad x > 0; \alpha_0 > 0, \tag{1}$$

Can anyone give explanation on how the gamma distribution forms a scale family.

(c) N$(1,\sigma^2)$ distributions does not form a scale family. How do I show this?


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