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Let $0<p_{i} <q_{i} <1$, where $i\in\{1,\ldots,n\}$.

I need to prove (I verified this numerically) that $$ B_{n} \le S_{n}$$ for any $n >k$, ($k$ is relatively small), where $$B_{n} =\frac{1}{2}\int\limits_{-\infty }^{-\infty } {\min\left( {f(x),\;g(x)} \right)} \;dx$$ $$F\sim N(\mu_{1} ,v_{1} )\;\text{ with }\;\operatorname{pdf}\,f,\quad G\sim N(\mu_{2} ,v_{2} )\;\text{ with }\;\operatorname{pdf}\,g$$ $$\mu_{1} =\frac{2}{n}\sum\limits_{i=1}^n {p_{i} } , \quad \mu_{2} =\frac{2}{n}\sum\limits_{i=1}^n {q_{i} } , \quad v_{1} =\frac{2}{n^{2}}\sum\limits_{i=1}^n {p_{i} (1-p_{i} )} , \quad v_{2} =\frac{2}{n^{2}}\sum\limits_{i=1}^n {q_{i} (1-q_{i} )}$$

and $$ S_{n} =\frac{1}{2}\int\limits_{x=-\infty }^\infty {\left[ {w_{1} (x)\int\limits_{y=-\infty }^x {h(y)\;dy} } \right]\;} dx+\frac{1}{2}\int\limits_{x=-\infty }^\infty {\left[ {w_{2} (x)\int\limits_{y=-\infty }^x {h(y)\;dy} } \right]\;} dx$$ $$W_{1} \sim N(\mu_{W1} ,v_{W1} )\;\text{ with }\;\operatorname{pdf}\,w_{1} ,\quad W_{2} \sim N(\mu _{W2} ,v_{W2} )\;\text{ with }\;\operatorname{pdf}\,w_{2}$$ $$H\sim N(\mu_{B} ,v_{B} )\;\text{ with }\;\operatorname{pdf}\,h$$

$$\mu_{W1} =\frac{4}{n}\sum\limits_{i=1}^n {(-p_{i}^{4}+2p_{i}^{3}-2p_{i} ^{2}+p_{i} )}$$ $$\mu_{W2} =\frac{4}{n}\sum\limits_{i=1}^n {(-q_{i}^{4}+2q_{i}^{3}-2q_{i} ^{2}+q_{i} )}$$ $$\mu_{B} =\frac{2}{n}\sum\limits_{i=1}^n {(2p_{i} q_{i}^{2}+2p_{i} ^{2}q_{i} -2p_{i}^{2}q_{i}^{2}-4p_{i} q_{i} +p_{i} +q_{i} )}$$

$$v_{W1} =\frac{4}{n^{2}}\sum\limits_{i=1}^n {\left( {p_{i} (1-p_{i} )-4(-p_{i}^{4}+2p_{i}^{3}-2p_{i}^{2}+p_{i} )^{2}} \right)}$$ $$v_{W2} =\frac{4}{n^{2}}\sum\limits_{i=1}^n {\left( {q_{i} (1-q_{i} )-4(-q_{i}^{4}+2q_{i}^{3}-2q_{i}^{2}+q_{i} )^{2}} \right)}$$ $$v_{B} =\frac{2}{n^{2}}\sum\limits_{i=1}^n {\left( {p_{i} (1+p_{i} )+q_{i} (1+q_{i} )-4p_{i} q_{i} } \right.} \left. {-2(2p_{i} q_{i}^{2}+2p_{i} ^{2}q_{i} -2p_{i}^{2}q_{i}^{2}-4p_{i} q_{i} +p_{i} +q_{i} )^{2}} \right)$$

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2  
The Horror. The Horror. –  Did Jul 15 '12 at 15:30
    
;) Sorry about that... I tried to frame it as succinctly as possible. –  Omri Jul 15 '12 at 15:33

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