NormalDistribution.stan

functions {
  // Define log probability density function
  real myNormal_lpdf(real y, real mu, real sigma) {
    return -log(2 * pi()) / 2 - log(sigma) 
           - square(mu - y) / (2 * sigma^2);
  }
}
data {
  int N;
  real y[N];
}
parameters {
  real mu;
  real<lower = 0> sigma;
}
model {
  // Priors
  mu ~ normal(0, 10);
  sigma ~ cauchy(0,10);
  // Likelihood
  for(n in 1:N){
    target +=  myNormal_lpdf(y[n] | mu, sigma);
  }
}
generated quantities{
  real y_ppc;
  {
    real x;
    x = uniform_rng(0, 1);
    // Phi is the probit function in Stan, the CDF of the standardised Normal N(0,1)  
    // inv_Phi is the quantile function of the standardised Normal N(0,1)
    y_ppc =  mu + sigma * inv_Phi(x);  
  }
}