What is Cumulative Distribution Function of this random variable? Suppose that we have $n$ independent random variables, $x_1,\ldots,x_n$ such that each $x_i$ takes value $a_i$ with success probability $p_i$ and value $0$ with failure probability $1-p_i$ ,i.e.,
\begin{align}
P(x_1=a_1) & = p_1,\  P(x_1=0)= 1-p_1 \\
P(x_2=a_2) & = p_2,\ P(x_2=0) = 1-p_2 \\                                                                                                                                                                                                                                   
& \vdots \\
P(x_n=a_n) & = p_n,\ P(x_n=0)=1-p_n  
\end{align}
where $a_i$'s are positive Real numbers.
What would be the CDF of the sum of these random variables? That  is, what would be  $P(x_1+\cdots+x_n\le k)$ ? and how can we find it in an efficient way?

This answer is an attempt at providing an answer to a previous version of the
question in which the $x_i$ were independent Bernoulli random variables with
parameters $p_i$.
$P\{\sum_{i=1}^n x_i = k\}$ equals the coefficient of 
$z^k$ in $(1-p_1+p_1z)(1-p_2+p_2z)\cdots(1-p_n+p_nz)$. This can be found by 
developing the Taylor series for this function. It is not much easier than grinding out the answer by brute force.
