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Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

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Minimum conditional expectation of complement of event given conditional expectation of event?

Suppose $X$ is a pdf over $[0,m]$ and $Y$ is a binary experiment on $X$ such that $P(Y=1|X)$ is continuous, and we have that $\mathbb{E}[X|Y=1] = \mu_y$ and $\mathbb{E}[X] < \mu_y$. Is it always the c …
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