probability density of the maximum of samples from a normalized uniform distribution Suppose
$$X_1, X_2, \dots, X_n\sim Unif(0, 1), iid$$
and suppose
$$\hat\theta = \max\{X_1, X_2, \dots, X_n\} / \sum_i^nX_i$$
How would I find the probability density of $\hat\theta$?
I know the answer if it's iid. But I don't know how to formalize the fact that the sum is iqual to 1.
a simiar question can be found here: 
probability density of the maximum of samples from a uniform distribution

I arrive here:
\begin{align}
P(Y\leq x)&=P(\max(X_1,X_2 ,\cdots,X_n)/\sum_i^nX_i\leq x)\\&=P(X_1/\sum_i^nX_i\leq x,X_2/\sum_i^nX_i\leq x,\cdots,X_n/\sum_i^nX_i\leq x)\\
&\stackrel{ind}{=} \prod_{j=1}^nP(X_j/\sum_i^nX_i\leq x )\\& \ \ \ \ \ 
\end{align}

Note that $$\hat\theta_n\sim \frac 1 {1+\sum_{i=1}^{n-1}U_i} $$
for i.i.d. standard uniforms $U_i$. Now see this and this
