# Is pdf from power family distribution an exponential family?

In my statistics class we've been going over exponential families and sufficiency, which deviates from what's in the textbook. As such, now that I need to solve problems about exponential families I don't have examples and explanations in the textbook to fall back on, and unfortunately the examples we go over in class are simple and not as complex as the problems I want to solve now.

My question is: is a power family distribution with parameters $\alpha$ and $\theta$ where $f(y|\alpha, \theta)=\frac{\alpha}{\theta^{\alpha}}y^{\alpha-1}$ for $0\leq y\leq\theta$, an exponential family?

I had thought that because the support $0\leq y\leq \theta$ depended on $\theta$ it couldn't be an exponential family, but a classmate told me this was incorrect. Could somebody tell me how I can get this pdf into the form $f(y, \theta)=h(y)c(\theta)e^{\sum w_i(\theta)t_i(y)}$ with the support containing $\theta$?

You're correct that this family of distributions with varying $\theta$ is not an exponential family; you can't factor the condition $y > \theta$ into something dependent only on $y$ and something only on $\theta$.
If you hold $\theta$ fixed, the resulting family with the single parameter $\alpha$ then is an exponential family (for any single $\theta$). This is analogous to the case for, say, a binomial distribution.