I want to evaluate the following expression which is related to Bernoulli trials:
For $i\in\{1,2,\cdots,n\},$ let $q_i\in[0,1]$ and $x_i\in\{0,1\}$. Furthermore, we know $q_i\geq q_{i+1}.$ Here, $q_i$ is the success probability of a Bernoulli trial and $x_i=1$ indicates a 'success'.
We define a joint distribution $f$ as follows $$f(q_1,\cdots,q_n)=A\prod^k_{i=1}q_i^{x_i}(1-q_i)^{1-x_i},~k<n.$$ Here, $A$ is a normalizing constant that makes $f$ a proper pdf.
What I want to calculate is the marginal distribution of $q_{k+1}.$ So, the object is to find the value of the following integration: $$\int f(q_1,\cdots,q_n)dq_1dq_2\cdots dq_{k}dq_{k+1}\cdots dq_{n}.$$ or equivalently, $$\int^1_{q_{k+1}}\cdots\int^{q_{k-1}}_{q_{k+1}}\int^{q_{k+1}}_0\cdots\int^{q_{n-2}}_0\int^{q_{n-1}}_{0}f(q_1,\cdots,q_n)dq_ndq_{n-1}\cdots dq_{k+2}dq_{k}\cdots dq_{1}.$$
Would there be any concise expression of the above multiple integration?