The following proposition stems from the proof of Neyman-Pearson lemma. The result is not quite intuitive, and is there any easier proof?
Proposition 1 Let
are two random variables. Consider
. Suppose there exists
satisfying
then
.
![\displaystyle \mathbb{E}[F I_{\{\hat a F<H\}}] = x,](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cmathbb%7BE%7D%5BF+I_%7B%5C%7B%5Chat+a+F%3CH%5C%7D%7D%5D+%3D+x%2C&bg=ffffff&fg=000000&s=0&c=20201002 "\displaystyle \mathbb{E}[F I_{\{\hat a F<H\}}] = x,")
The following proposition stems from the proof of Neyman-Pearson lemma. The result is not quite intuitive, and is there any easier proof?
Proposition 1 Let
then
