For my study I need to provide a data set that includes three weibull distributed samples with the same shape, but different scale.
In the best case, I want to do a transformation onto one sample, to get as a result a sample with the same shape, but another scale. Is this possible?
(Additional:
I assume, I need to find an equation, that I can solve for the scale, and then use it as transformation, but I am a little lost, to find the equation to start with. I have tried the PDF, but I am not able to solve it for the scale (nor is the software I used).
The lecturer does not allow to generate a random sample and I do not want to lie. Besides I assume I am expected to learn something by this, but I am not able to see it until now.)
If $X\sim Weib(\lambda=1,k)$ we have $$\mathbb P(X<t)=1-e^{-t^k}.$$
Your goal, if I have well understood is to arrive to a generic Weibull distribution, where, $$\mathbb P(f(X)<t)=1-e^{-(t/\lambda)^k}$$ applying a function $f$ on $X$. Now, consider that for any random variable $Y$ with distribution $F_Y$ and any $a>0$ we have $$F_{aY}(t)=\mathbb P(aY<t)=\mathbb P(Y<t/a)=F_Y(t/a)$$