I have: $$p_{\text{sigmoid}_i} =\frac{e^{a+b{x_i}}}{1+e^{a+b{x_i}}}$$ where $x_1, \ldots, x_n$'s are generated from a normal distirbution with mean $\mu_0$ and $x_{n+1},\ldots,x_{2n}$ are generated from $\mu_1$. The $\sigma$ for both samples is the same. ($a$ and $b$ are some known parameters.)
Using $p$, I create $n$ Bernoulli experiments with $p = p_\text{sigmoid}$.
After getting the results of Bernoulli, by using Neyman-Pearson test, I need to discriminate $$\mathcal{H_0} = \Pr(m = m_0\mid y=1)$$ and $$\mathcal{H_1} = \Pr(m = m_1 \mid y=1)$$ between the samples created from the Bernoulli which originally created from $\mu_1$ and $\mu_0$
My question is what is the sum of sigmoid of normal distribution? (with mean and variance as it's parameters so I can use it in log-likelihood ration)