Consider \(\displaystyle N\) random variables \(\displaystyle X_{n}\) each following a Bernoulli distribution \(\displaystyle B(r_{n})\) with \(\displaystyle 1 \geq r_{1} \geq r_{2} \geq ... \geq r_{N} \geq 0\). If we make following assumptions of sets \(\displaystyle A\) and \(\displaystyle B\):
(1) \(\displaystyle A \subset I \) and \(\displaystyle B \subset I\) with \(\displaystyle I=\{1,2,3,...,N\}\)
(2) \(\displaystyle |A \cap I_{1}| \geq |B \cap I_{1}|\) with \(\displaystyle I_{1}=\{1,2,3,...,n\}, n<N\)
(3) \(\displaystyle |A|=|B|=n\)
Do we have \(\displaystyle \mathbb{E}(\Sigma_{a\in A} X_{a}) \geq\mathbb{E}(\Sigma_{b\in B} X_{b})\)?
To avoid confusion, \(\displaystyle \mathbb{E}\) means expected value.
Thanks!
(1) \(\displaystyle A \subset I \) and \(\displaystyle B \subset I\) with \(\displaystyle I=\{1,2,3,...,N\}\)
(2) \(\displaystyle |A \cap I_{1}| \geq |B \cap I_{1}|\) with \(\displaystyle I_{1}=\{1,2,3,...,n\}, n<N\)
(3) \(\displaystyle |A|=|B|=n\)
Do we have \(\displaystyle \mathbb{E}(\Sigma_{a\in A} X_{a}) \geq\mathbb{E}(\Sigma_{b\in B} X_{b})\)?
To avoid confusion, \(\displaystyle \mathbb{E}\) means expected value.
Thanks!