# probability – Scale parameter definition

I have seen two definitions of scale parameter of a distribution: Let $$X$$ be random variable and $${{P_{theta, X }: theta in Theta subset mathbb{R}}}$$ be a parametric family of distribution of $$X$$ indexed by $$theta$$

$$1)~~ theta text{ is a scale parameter iff} rightarrow text{ the distribution of } cX text{ is in } {{P_{theta, X }: theta in Theta subset mathbb{R}}} text{ for every positive constant c}$$

For example Let $$X$$ ~ $$LogNormal(theta, 1)~, theta in (-infty, infty)$$ then $$cX$$ ~ $$LogNormal(theta^*,1)$$ where $$theta^* = theta + ln(c) in (-infty, infty)$$. With this definition $$theta$$ is a scale parameter.

For the second definition it is required that $$theta$$ is positive:

$$2) ~ theta > 0 text{ is a scale parameter } iff rightarrow text{ the distribution of } X/theta text{ is independent of } theta$$

If $$X$$ ~ $$LogNormal(theta, 1)~, theta in (0, infty)$$ (We restrict the parameter space) then $$X/theta$$~ $$LogNormal(theta + ln(1/theta), 1)$$. With this definition $$theta$$ is not a scale parameter.

I was wondering which of these two is the correct definition of a scale parameter? Or it depends on the book and the author?