Is there a well-known PDF with the following description $f_U\left(u\right) = \frac{\alpha}{\beta^{\alpha}} u^{\alpha-1} $ where $0 \leq u \leq \beta$

Question

Is there a well-known PDF with the following description

$$f_U\left(u\right) = \frac{\alpha}{\beta^{\alpha}} u^{\alpha-1} \quad \text{where} \quad 0 \leq u \leq \beta$$

I was working with a problem where I have shown that

$$\int_0^\beta \frac{\alpha}{\beta^{\alpha}} u^{\alpha-1} = 1$$

Hence, $f_U\left(u\right)$ can be a legitimate PDF for some random variable called $U$. Now, I am wondering if there is any well-known or standard name for this type of distribution? I've done a quick search and the closest thing I could find was Gamma PDF, but its definition has some fundamental discrepancies with the above expression.

So, I am just wondering if there is any specific and stablished name for describing such PDFs?

Answer 1

This seems to be the PDF of $U=\beta X$ where $X \sim \text{Beta}(\alpha, 1)$. The support of $U$ is $[0,\beta]$ while the support of $X$ is $[0,1]$, and [ignoring normalizing constants] the main term of the PDF is $u^{\alpha - 1}$ which appears in the beta PDF.

Answer 2

If $\alpha$ is integer, $f_U$ can be interpreted as the pdf of

$$\max(X_1,X_2,\cdots X_{\alpha})$$

where the $X_k$s are iid (independant uniformly distributed) on $[0,\beta]$.

It is the member of the wider family of the so-called "order-statistics".