I had a question of two parts. I solved the first part but I am stuck on the second. Any hints or partial solutions would be greatly appreciated.
a)$ X_1,....X_n$ are uniform iid on the interval $(0,\theta)$. If $Y=max(X_1,....X_n)$ find the pdf of Y? I worked the pdf to be $n \dfrac{\bigl(\dfrac{x}{\theta}\bigr )^{n-1}}{\theta}$
b) Construct a random variable $T = g(Y;\theta)$ such that T has the same distribution as Y for all values of $\theta$ . Such a random variable T will then be a pivot? I am honestly not sure where to start with this one?
A pivot is a random variable $T=g(Y;θ)$ such that $T$ has the same distribution for all values of $θ$ (and not what is written in the question). If one notes that $X$ uniform on $(0,\theta)$ means that $X/\theta$ is uniform on $(0,1)$, one can guess that $T=Y/\theta$ is indeed a pivot. Can you check this?
More generally, if the density of some random variable $Z$ depending on $\theta$ has the form $f_Z(z\mid\theta)=g(z/\theta)/\theta$ for some function $g$ then $g$ is a PDF and $Z/\theta$ is a pivot.