Let $X= \sum_{i=1}^{N} X_i$, where $X_i \sim Bernoulli(p_i)$. Let $Y= \sum_{i=1}^N Y_i$, where $Y_i \sim Bernoulli(p_i+ \delta)$ for some $0 \leq \delta \leq 1- \max_i p_i$.
Can we prove $P(X \leq k) \geq P(Y \leq k), \forall k \in \{0, \dots, N\}$?
Comparing Cdf of Poisson Binomial distribution for different parameters
Rakshith
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