The outputs $Y_{1}, \ldots, Y_{n}$ of a particular process are positive and are believed to be related to $p$-vectors of covariates $x_{1}, \ldots, x_{n}$ according to the following model

$$\log \left(Y_{i}\right)=\mu+x_{i}^{T} \beta+\varepsilon_{i}$$

In this model $\varepsilon_{i}$ are i.i.d. $N\left(0, \sigma^{2}\right)$ random variables where $\sigma>0$ is known. It is not possible to measure the output directly, but we can detect whether the output is greater than or less than or equal to a certain known value $c>0$. If

$$Z_{i}= \begin{cases}1 & \text { if } Y_{i}>c \\ 0 & \text { if } Y_{i} \leqslant c\end{cases}$$

show that a probit regression model can be used for the data $\left(Z_{i}, x_{i}\right), i=1, \ldots, n$.

How can we recover $\mu$ and $\beta$ from the parameters of the probit regression model?