The assumptions in the derivation of normal distribution using a dart board example?

Question

The derivation could be found here:

https://math.stackexchange.com/questions/384893/how-was-the-normal-distribution-derived

or here https://www.youtube.com/watch?v=cTyPuZ9-JZ0

They assume that (1) the pdf f(x), f(y) are independent from each other and (2) the pdf of a point g(x,y) only depends on the distance of the point to the bull's eye.

There are definitely examples of pdfs that do not satisfy both (1) and (2). For example, imagine a discrete example where the dart landing at (2,1) or (1,2) both have a probability of 0.4 and the dart landing at (2,2) has a probability of 0.2. This satisfy the assumption (2) that the probability depends only on the distance but not (1). What kind of pdfs can satisfy both (1) and (2)?

Answer 1

Joint pdfs $f_{X,Y}(x,y)$ that have circular symmetry about the origin (property 2 in the OP's description) exist in profusion, and they all have the property that if we think of $X$ and $Y$ in terms of $R=\sqrt{X^2+Y^2}$ and $\Theta = \arctan\left(\frac YX\right)$, then $R$ and $\Theta$ are independent random variables, and $\Theta \sim U[0,2\pi)$. But, properties 1 (independent $X$ and $Y$) and 2 (circular symmetry about the origin) are harder to satisfy.

It is shown in Sheldon Ross's A First Course in Probability, Prentice-Hall (see Example 2e in Chapter 6.2) that, under the slightly more onerous constraints that the marginal pdfs $f_X(x)$ and $f_Y(y)$ are differentiable functions of $x$ and $y$ respectively and that $f_{X,Y}(x,y)$ is differentiable with respect to $x$ and also with respect to $y$, the only pdfs $f_X$ and $f_Y$ that satisfy both properties 1 and 2 are zero-mean Gaussian pdfs with identical variances.