# Generating samples of a Gaussian distribution with independent components above a hypersurface

I recently asked this question (link here) regarding generating samples of a Gaussian vector with independent components above a hyperplane. Before getting an answer, I had already performed a simple simulation in 2-D: Let $Z=(Z_1,Z_2)$ be a Gaussian vector with $N(0,1)$ independent components. I tried generating samples of $Z$ such that $Z_1>4$ (i.e. my hyperplane) using what I thought was an intuitive approach:

1) Generate a sample of $Z_2 \sim N(0,1)$.

2) Since $Z_1,Z_2$ are independent, I just had to generate a sample of $Z_1 > 4$.

3) Then this gives me a sample of $Z = (Z_1,Z_2)$.

Now, I am interested in generating samples of a Gaussian vector that lie above a hypersurface. For simplicity, let's consider the 2-D example in which I want to generate $Z$ (same vector above) such that $Z_1 > Z_2^2 + 3$. I adapted the "intuitive" approach I pursued above as follows: