probit regression, in which a binary variable Yi is described in terms of an explanatory variable x via the following latent variable model: Zi = Bxi + ci Yi = 6(c,00) (Zi), where 3 and c are unknown coefficients, 61, . .., En ~ i.i.d. normal(0, 1) and o(c,) (2) = 1 if z > c and equals zero otherwise. a) Assuming 3 ~ normal(0, 73) obtain the full conditional distribution p( B y, x, z, c). b) Assuming c ~ normal(0, T2 ), show that p(cy, x, z, 3) is a constrained normal density, i.e. proportional to a normal density but constrained to lie in an interval. Similarly, show that p(zi y, x, z_, B, c) is pro- portional to a normal density but constrained to be either above c or below c, depending on yi