Let $\left\{X_{{1}}, X_{{2}}, \ldots, X_{k}\right\}$ denotes Bernoulli random variables with $\quad\left[p_{1}, p_{2}, \ldots, p_{k}\right]$ as respective parameters. Let
$S=a_1X_{{1}}+a_2X_{{2}}+\ldots+a_kX_{k}$.
where $a_i > 0$ and $a_i \neq 0, \forall i \in {1,2,\ldots,k}$
Note that all $p_i$ values are closer to $0$ (but not equal to) and they are different. Moreover, $k$ is not too large to invoke asymptotic approximations or not too small.
What are some suitable weights $a_i$'s such that the distribution of $S$ has a closed form?
Note that the trivial solution of $a_i=constant$ is not useful for me. Some possibility of weights I could consider are as follows:
- $a_i=i \implies S=X_{{1}}+2X_{{2}}+\ldots+kX_{k}$ . Does the distribution of S have a closed form in this case.?
- $a_i =q^{i-1} \implies S=X_{{1}}+q^1X_{{2}}+\ldots+q^{k-1 }X_{k}$ . This has geometric weights.
- $a_i=1/i \implies S=X_{{1}}+(1/2)X_{{2}}+\ldots+(1/k)X_{k}$
I am flexible with choosing any weight sets as long as they are all positives and unique.