Given $T_1,\ldots, T_n$ are i.i.d random variables with pdf $\ f_{T}(t|\theta) = (\theta+1)t^{\theta}I_{(0,1)}(t)$. Find the one-dimensional sufficient statistics.
My attempt: We have: $f(\overline{t}|\theta) = (\theta+1)^n(t_1\ldots t_n)^\theta\prod_{i=1}^{n} I_{(0,1)}(t_i) $. Thus if we let $t_1\ldots t_n = t$, then we have: $g(t|theta) = (\theta+1)^nt^\theta$ and $h(\overline{t}) = I_{(0,1)}(t_{(1)})$ where $\ t_{(1)} = min\left\{t_1,t_2,\ldots, t_n\right\}$. Therefore, by Factorization Theorem, the one-dimensional sufficient statistic is $\ t_1\ldots t_n$
My question: Is the solution above correct? I'm skeptical a bit since it's quite simple.