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Suppose that $X_1, X_2, \cdot \cdot \cdot \ , X_n$ is a random sample from a continuous distribution with pdf $f_X(x;\theta) = \theta x^{\theta-1}$, for $\ 0\leq x \leq 1$. Show that $W=\prod_{i=1}^n X_i$ is a sufficient statistic for $\theta.$ Is the maximum likelihood estimator for $\theta$ a function of $W$?

Here's what I did:

$$P(X_1=x_1,...,X_n=x_n|W=w)=\cfrac{P(X_1=x_1,...,X_n=x_n,\prod_{i=1}^nX_i=\prod_{i=1}^nx_i)}{P(\prod_{i=1}^nX_i=\prod_{i=1}^nx_i)} \stackrel{redundancy}= \cfrac{P(X_1=x_1,...,X_n=x_n)}{P(\prod_{i=1}^nX_i=\prod_{i=1}^nx_i)}=\cfrac{\theta^n(\prod_{i=1}^nx_i)^{\theta-1}}{P(\prod_{i=1}^nX_i=\prod_{i=1}^nx_i)}$$

Here I am stuck. I am having difficulty breaking up the probability in the denominator. Thanks for the help.

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    $\begingroup$ Your distribution is $\mathsf{Beta}(\theta, 1).$ Wikipedia has a very long article on beta distributions (showing lots of pretty pictures, at least), with a discussion on MLEs (hence sufficiency) more than halfway down. Possibly useful. $\endgroup$ – BruceET Apr 14 '20 at 1:52
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    $\begingroup$ The analysis of $\log(W)$ is much easier. $\endgroup$ – whuber♦ Apr 14 '20 at 2:57
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    $\begingroup$ This question and its answers can be found in multiple posts on this site. But you cannot work with probabilities here (it is a continuous distribution). Do you know Factorization theorem? $\endgroup$ – StubbornAtom Apr 14 '20 at 7:17
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    $\begingroup$ Sufficiency is mentioned here and MLE is mentioned here for example. These do not discuss your exact questions of course but they contain the answers which you would have to work out on your own. You might want to add the self-study tag to your question and read the tag wiki. $\endgroup$ – StubbornAtom Apr 15 '20 at 20:38
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    $\begingroup$ @Xi'an heck yeah it does $\endgroup$ – jeremy909 Jun 24 '20 at 16:18