 Solution

The standard normal distribution function $\Phi$ givesThe observations are independent Bernoulli random variables, because the original errors are independent. Thus the probit regression has intercept $\alpha =(\mu -\log c)/\sigma$ and coefficient vector $\gamma =\beta /\sigma$. With the specified known $\sigma$ and $c$, recovery isThe scale would not be recoverable from threshold data alone if $\sigma$ were unknown.