How does multiplication by a constant affect a Gumbel random variable

Question

Suppose $X$ is a Gumbel (Type-1 extreme value) random variable with shape and scale parameters given by ($\mu$, $\sigma$). What is the distribution of $cX$, where $c$ is a constant?

Answer 1

The distribution will still be a Gumbel distribution but with parameters $(c\mu,c\sigma)$.

That is the case for either of the usual Gumbel Type 1 distributions. For $Y=cX$ those have density functions:

$$\frac{e^{\frac{\alpha c-y}{\beta c}-e^{\frac{\alpha c-y}{\beta c}}}}{\beta c}$$ and $$\frac{e^{\frac{y-\alpha c}{\beta c}-e^{\frac{y-\alpha c}{\beta c}}}}{\beta c}$$