Let \(y_{i}^{*}=\beta_{0}+\boldsymbol{\beta}^{\top} \mathbf{x}_{i}+\varepsilon_{i}\), where \(\varepsilon_{i} \sim N(0,1)\) (a standard normal with mean 0 and variance 1 ) and \(y_{i}\) is determined by \(y_{i}^{*}\) as an indicator for whether this latent variable is positive, i.e.,

\[y_{i}= \begin{cases}1 & \text { if } y_{i}^{*}>0, \text { i.e., }-\varepsilon_{i}<\beta_{0}+\mathbf{x}_{i}^{\top} \beta \\ 0 & \text { if otherwise. }\end{cases}\]

Show that \(\operatorname{Pr}\left(y_{i}=1 \mid \mathbf{x}_{i}\right)=\Phi\left(\beta_{0}+\mathbf{x}_{i}^{\top} \boldsymbol{\beta}\right)\), where \(\Phi\) is the CDF of standard normal.