In Frank Harrell's Regression Modeling Strategies, he states:
The ordinary linear regression model is:
$$C(Y|X)=E(Y|X)=X\beta$$
and given $X$, $Y$ has a normal distribution with mean $X\beta$ and constant variance $\sigma^2$. The binary logistic regression model is:
$$C(Y|X)=Prob(Y=1|x)=(1+exp(-X\beta))^{-1}$$$$C(Y|X)=\textrm{Prob}(Y=1|x)=(1+\exp(-X\beta))^{-1}$$
How is this formula $(1+exp(-X\beta))^{-1}$$(1+\exp(-X\beta))^{-1}$ derived? I have tried looking at his cited sources but it is still not clear to me.
How do we go from $C(Y|X)=E(Y|X)=X\beta$ to $Prob(Y =1|X)=(1+exp(-X\beta))^{-1}$$\textrm{Prob}(Y =1|X)=(1+\exp(-X\beta))^{-1}$?
