Prove that if [operatorname{Pr}left(y_{i}=1 mid mathbf{x}_{i} ight)=Phileft(beta_{0}+mathbf{x}^{top} boldsymbol{beta} ight), quad operatorname{Pr}left(y_{i}=0 mid mathbf{x}_{i} ight)=Phileft(alpha_{0}+mathbf{x}^{top} boldsymbol{alpha} ight)] where

Prove that if

\[\operatorname{Pr}\left(y_{i}=1 \mid \mathbf{x}_{i}\right)=\Phi\left(\beta_{0}+\mathbf{x}^{\top} \boldsymbol{\beta}\right), \quad \operatorname{Pr}\left(y_{i}=0 \mid \mathbf{x}_{i}\right)=\Phi\left(\alpha_{0}+\mathbf{x}^{\top} \boldsymbol{\alpha}\right)\]

where \(\Phi\) is the CDF of standard normal, then \(\beta_{0}=-\alpha_{0}\) and \(\boldsymbol{\beta}=-\boldsymbol{\alpha}\).