# From Uniform to 2D gaussian

I have a 2D uniform distribution. I would like to assign probability-weights in the distribution, according to the standard deviation to convert the distribution into a 2D gaussian.

I know how to do it in the 1D case, by splitting the points from -1 to 1 sigma, etc and assigning the probability as a weight but I haven't found a way to combine the probabilities from the 2 planes (eg at 1 sigma in x and 6 sigma in y)

## 1 Answer

The Gaussian distribution is separable. Apply your transformation to each coordinate separately and you will get a 2D Gaussian.

If $G(x)$ is a 1D Gaussian, then $G(x) G(y)$ is a 2D Gaussian.

Thus, transform the distribution first according to one coordinate, creating a distribution that has normal distribution in one direction and uniform in the other. Then transform it again according to the other coordinate.

• A point between 0-1 sigma in x has Px=0.34 for a normal gaussian. The same point is in 2-3 sigma in y and has a Py = 0.02. How can I combine both probabilities and assign only one value to each point? – Uno123 Apr 9 '18 at 13:46
• By multiplication. See the edit of the answer. – Cris Luengo Apr 9 '18 at 14:03