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For the Box-Muller transform, draw independent $U,V$ from the uniform distribution on $(0,1)$ and setThe radial density is $r{e}^{-r^2/2}$ for $r>0$, with an independent uniform angle. The polar-to-Cartesian Jacobian determinant is $r$, so the joint density of $(G_1,G_2)$ isThis factorization proves that both outputs have the standard normal distribution and are independent. Repeating the Box-Muller transform with fresh independent uniform pairs gives independent normal outputs; discard one extra output if the desired sample size is odd.

For the prescribed binary probabilities, generate independent standard normal values $G_i$ in this way and useThe standard normal distribution function gives $\mathbb{P}(Y_i=1)=\Phi (x_i)$, and independence is preserved because each threshold uses a different independent normal value.

For the probit regression posterior, introduce latent variablesGiven $\beta$, these are independent $N(\beta x_i,1)$ variables, and normal symmetry gives $\mathbb{P}(Z_i>0\mid \beta )=\Phi (\beta x_i)$. Thus this data augmentation has exactly the observed binary likelihood. With the specified normal prior distribution, the augmented joint posterior is proportional toThe latent-normal Gibbs sampler for probit regression alternates two blocks. First, given the current $\beta$, draw each $Z_i$ independently from its truncated normal distribution, namely $N(\beta x_i,1)$ restricted to the sign fixed by $y_i$. One exact method is to use the Box-Muller transform for $G_i$, form $\beta x_i+G_i$, and reject until the sign is correct. The probability of success is positive at every finite parameter value, so the method is valid, although it can be slow for a rare sign.

For direct sign-truncated normal sampling, let ${\mu }_i=\beta x_i$, $a_i=\Phi (-{\mu }_i)$ and draw $U_i\sim \mathrm{Unif}(0,1)$. An inverse transform sampling implementation isUse suitable tail or survival-function evaluations when floating-point probabilities approach zero or one; the rejection construction remains a valid alternative.

Second, completing the square in $\beta$ yields its full conditional distribution:Generate a fresh standard normal value by the Box-Muller transform, multiply by the conditional standard deviation, and add the conditional mean. Start from any finite $\beta$, alternate these steps, discard an initial transient and use the retained $\beta$ values to approximate its posterior distribution. These are dependent Markov chain Monte Carlo samples, rather than independent posterior draws. Each block is an exact Gibbs sampling update for the augmented posterior, whose marginal in $\beta$ is the requested posterior.