k,k}
\ = \
\prod_{k=1}^{K-1}
z_k
\,
(1 - z_k)
\
\left(
1 - \sum_{k&#39;=1}^{k-1} x_{k&#39;}
\right)
.
\]</span></p>
<p>Thus the transformed variable <span class="math inline">\(Y = f(X)\)</span> has a density given by</p>
<p><span class="math display">\[
p_Y(y)
= p_X(f^{-1}(y))
\,
\prod_{k=1}^{K-1}
z_k
\,
(1 - z_k)
\
\left(
1 - \sum_{k&#39;=1}^{k-1} x_{k&#39;}
\right)
.
\]</span></p>
<p>Even though it is expressed in terms of intermediate values <span class="math inline">\(z_k\)</span>,
this expression stil