Comparing Cdf of Poisson Binomial distribution for different parameters

Question

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 $0 \leq \delta \leq 1- \max_i p_i$. All considered random variables are independent.

Can we prove $P(X \leq k) \geq P(Y \leq k), \forall k \in \{0, \dots, N\}$?

Answer 1

Yes. Let $Z_i\sim Bernoulli(\frac{\delta_i}{1-p_i})$ (with independence) and $Y_i=\min\{1,X_i+Z_i\}$. Then $Y_i\sim Bernoulli(p_i+\delta_i)$. But since $Y_i\ge X_i$ for all $i$, you have $\{Y\le k\}\subset\{X\le k\}$ for all $k$.