How do I use Monte Carlo's simulation to approximate part of the ellipsoid X^2/2+y^2/4+z^2/8=16, that lies in the first octant x>0,y>0,z>0?
Rejection sampling:
Generate a point $\displaystyle (x,y,z)$ uniformly distributed over $\displaystyle [0,\sqrt{32}]\times[0,8]\times[0,\sqrt{128}]$ accept this point if:
$\displaystyle \frac{x^2}{2}+\frac{y^2}{4}+\frac{z^2}{8}\le 16$
If not accepted repeat until an acceped point is found
The resultant $\displaystyle (x,y,z)$ is uniformly distributed over the required ellipsoid.
The efficiency of this process is $\displaystyle \approx 0.507$ which means the volume of the ellipsoid in the first octant is $\displaystyle \approx 0.507 $ of the volume of $\displaystyle [0,\sqrt{32}]\times[0,8]\times[0,\sqrt{128}]$
CB