Suppose the distribution of X for subjects having Y=1 is normal $\displaystyle N(\mu_1,\sigma)$, and suppose the distribution of X for subjects having Y=0 is normal $\displaystyle N(\mu_0,\sigma)$. Then why is $\displaystyle P(Y=1|x)$ a logistic regression curve?

And why $\displaystyle \beta=(\mu_1-\mu_0/\sigma^2)$ ?