I have a probability function $P$ depends on a standard normal random variable $\eta$. Given a random draw of $\eta^r$ from the standard normal distribution ($\eta\~N(0,1)$), I can get a probability $P(\eta^r)$. I want to know if the following formula has its range, or could be calculated directly. $$E(\sum_r\frac{1}{R}P(\eta^r)(1-P(\eta^r))\eta^r)$$. I know that there is a bound for the probability $0=<P(\eta^r)(1-P(\eta^r))<=0.25\ \forall\eta^r$, and would this boundary of the multiplication of the probability help? $$P=\frac{e^{A_j+B_j\eta}}{\sum_je^{A_j+B_j\eta}}\ ,B_i>0 \ \forall i $$

Thanks a lot!


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