Iam a bit unsure if the distribution of the canonical statistic below is correct ? I suspect i would need idenpendent variables to show the correct answer.

Let $Y_1,Y_2,...Y_n$ be a sample form a distribution on the unit interval. With density given by

$$f(y:\alpha) = \alpha y^{\alpha -1}$$

Show that $T = ln (\prod_{i=1}^{i=n} Y_i)$ belongs to the exponential family.

$$P(T \leq t ) = P(ln (\prod_{i=1}^{i=n} Y_i) \leq ln (\prod_{i=1}^{i=n} y_i)) = P(\prod Y_i \leq \prod y_i) $$


The distribution $\prod Y_i $ has distribution function $F(\boldsymbol{y}) = (\prod y_i)^{\alpha}$ -assuming $\{Y_i\}$ are independent I dont know how to do it otherwise...?

and its derivative is $\alpha(\prod y_i)^{\alpha-1}$ which therefore is the density of T.

Therefore the its density can be written as;

$$f(\boldsymbol{y}:\alpha) = \alpha e^{(\alpha-1)ln(\prod y_i)}$$

which proves that T is in the exponential family