What is the name of the distribution and/or distribution function of random variable $X$?
$X=a_1 x_1+a_2 x_2+...+a_n x_n$
$x_1, ..., x_n$ are independent binomial variables (Bernoulli). $x_i=1$ with probability $p$ and $0$ else. $a_1, \dots, a_n$ are given deterministic integers (weights) $\geq 1$.
How do I show that $P(X>\frac{1}{2}\sum a_i)+\frac{1}{2}P(X=\frac{1}{2}\sum a_i) \geq p$ for $p \geq \frac{1}{2}$?
This is not a Poisson binomial distribution. It is a generalized Poisson binomial but with the nice property that $p$ is constant. This question gives an almost identical problem.
In the spirit of the answer to this question I can calculate probabilities as $P(s_k)=(1-p)^n \sum_{i_1,...,i_m |\sum a_{i_k}=s_k} \Bigl( \frac{p}{1-p}\Bigr)^{\sum i_k}$. However, this does not seem to get me closer to the proof.
It might also be possible to cleverly get the desired proof without a complete distribution function.