How is $\theta$, the polar coordinate, distributed when $(x,y) \sim U(-1,1) \times U(-1,1)$ vs. when $(x,y) \sim N(0,1)\times N(0,1)$?

Question

Let the Cartesian $x,y$ coordinates of a random point be selected s.t. $(x,y) \sim U(-10,10) \times U(-10,10)$.

Thus, the radius, $\rho = \sqrt{x^2 + y^2}$, isn't uniformly distributed as implied by $\rho$'s p.d.f.

Nonetheless I would expect $\theta = \arctan{\frac{y}{x}}$ to be almost uniform, excluding artifacts due to the 4 leftovers at the edges:

Following are the grafically calculated probability density functions of $\theta$ and $\rho$:

Now if I let $x,y$ be distributed s.t. $x,y \sim N(0,20^2)\times N(0,20^2)$ then $\theta$ seems uniformly distributed:

Why $\theta$ isn't uniform when $(x,y) \sim U(-10,10) \times U(-10,10)$ and is uniform when $x,y \sim N(0,20^2)\times N(0,20^2)$?

The Matlab code I used:

number_of_points = 100000;
rng('shuffle')

a = -10;
b = 10;
r = (b-a).*randn(2,number_of_points);
r = reshape(r, [2,number_of_points]);
I = eye(2);
e1 = I(:,1); e2 = I(:,2);
theta = inf*ones(1,number_of_points);
rho = inf*ones(1,number_of_points);

for i=1:length(r(1,:))
    x = r(:,i);
    [theta(i),rho(i)] = cart2pol(x(1),x(2));        
end

figure
M=3;N=1; bins = 360;
subplot(M,N,1); 
histogram(rad2deg(theta), bins)
title('Polar angle coordinate p.d.f');

subplot(M,N,2); 
histogram(rho, bins);
title('Polar radius coordinate p.d.f');

subplot(M,N,3); 
histogram(r(:));
title('The x-y cooridnates distrbution (p.d.f)');

Substituting the 3rd line: r = (b-a).*randn(2,number_of_points); with r = (b-a).*randn(2,number_of_points) +a ; will change the distribution of $(x,y)$ from normal to uniform.

Answer 1

You're referring to a transformation from a pair of independent variates $(X,Y)$ to the polar representation $(R,\theta)$ (radius and angle), and then looking at the marginal distribution of $\theta$.

I'm going to offer a somewhat intuitive explanation (though a mathematical derivation of the density does essentially what I describe informally).

Note that if you scale the two variables, X and Y by some common scale (e.g. go from U(-1,1) to U(-10,10) or from N(0,1) to N(0,20) on both variables at the same time) that makes no difference to the distribution of the angle (it only affects the scale of the distribution of the radius). So let's just consider the unit cases.

First consider what's going on with the uniform case. Note that the distribution is uniform over the unit square, so that the probability density in a region that's contained within $[-1,1]^2$ is proportional to the area of the region. Specifically, look at the density associated with an element of angle, $d\theta$ near the horizontal (near angle $\theta=0$) and on the diagonal (near angle $\theta=\pi/4$):

Clearly the probability element $df_\theta$ (i.e. area) corresponding to an element of angle ($d\theta$) is larger when the angle is near one of the diagonals. Indeed consider inscribing a circle inside the square; the area spanned by a given tiny angle within the circle is constant, and then the part outside the circle grows as we approach the diagonal, where it at its maximum.

This completely accounts for the pattern you see in the simulations.

Indeed, we can see that the density must be proportional to the length of the segment from the center of the square to its edge; simple trigonometry is sufficient to derive the density from there and then it's easy to find the constant required to make the density integrate to 1.

[Edit: added this next bit to discuss the radius, since the question has changed since my original answer.]

Note that if we had a uniform distribution over the unit circle (i.e. the one we inscribed in the square before) then the density of the radius for that would be proportional to the radius (consider the area of a small annular element of width $dr$ at radius $r$ - i.e. between $r$ and $r+dr$ - has area proportional to $r$). Then as we pass outside the circle, new annular regions with larger radius only get density contributions from the part in the square, so the density decreases (initially quite rapidly, then more slowly) between $1$ and $\sqrt 2$. (Again, fairly simple geometrical notions are sufficient to get the functional form of the density if it is needed.)

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By contrast, if the joint distribution is rotationally symmetric about the origin then the probability element at some angle doesn't depend on the angle (this is essentially a tautology!). The bivariate distribution of two independent standard Gaussians is rotationally symmetric about the origin:

(code for this image based on Elan Cohen's code here but there's a nice alternative here, and something between the two here)

Consequently the volume contained in some angle $d\theta$ is the same for every $\theta$, so the density associated with the angle is uniform on $[0,2\pi)$.

[The polar trick typically used for integrating the normal density over the real line can be used to figure out that the density of the squared radius is negative exponential, and from there the density of the radius is simple to identify by a simple transformation argument from the distribution function]

Answer 2

To complete the fairly good answers given by Glen and Michael, I'll just compute the density of $\theta$ when the distribution of $(X,Y)$ is uniform on the square $[-1,1]\times[-1,1]$. This uniform density is $1 \over 4$ on this square, $0$ elsewhere -- that is, the probability of sampling a point in a given region of the square is $1 \over 4$ the area of this region.

The region of interest for our question is the red sector on this drawing:

It’s a triangle delimited by the angle $\theta$ and $\theta + d\theta$. The probability of sampling a point in this triangle is the probability of sampling a value between $\theta$ and $\theta + d\theta$ — which is the density of $\theta$.

I’ll make the computation for $\theta\in \left[ -{\pi \over 4}, {\pi\over 4}\right]$ -- the whole density can be obtained by extending it by $\pi\over 2$ periodicity.

Elementary trigonometry show that the lower side has length $1\over \cos\theta$. The upper size has length $$ {1\over \cos(\theta + d\theta)} = {1\over \cos\theta} + {\sin \theta \over \cos^2\theta} d\theta.$$ (We’ll see that the precise value of the derivative doesn’t really matter here!)

Now the area of a triangle with two sides of lengthes $a$ and $b$ forming an angle $\alpha$ is ${1\over 2}ab\sin\alpha$, hence in our case $$ {1\over 2} \left({1\over \cos\theta} \right) \left({1\over \cos\theta} + {\sin \theta \over \cos^2\theta} d\theta\right) \sin d\theta = {d\theta \over 2\cos^2 \theta}$$ (we neglect higher powers of $d\theta$ and use $\sin d \theta= d\theta$).

Thus the density of $\theta$ is $$ {1\over 8 \cos^2 \theta}$$ for $\theta$ in $\left[-{\pi\over 4},{\pi\over 4}\right]$, and is $\pi\over 2$ periodic.

Verification:

x <- runif(1e6, -1, 1)
y <- runif(1e6, -1, 1)
hist( atan2(y,x), freq=FALSE, breaks=100)
theta <- seq(-pi, pi, length=500)
lines(theta, 0.125/cos((theta + pi/4)%%(pi/2) - pi/4)**2, col="red" )

Answer 3

I will answer the question about the normal case leading to the uniform distribution. It is well known that if $X$ and $Y$ are independent and normally distributed the contours of constant probability density is a circle in the $x-y$ plane. The radius $R= \sqrt{X^2+ Y^2}$ has the Rayleigh distribution. For a good discussion of this, see the Wikipedia article titled Rayleigh distribution.

Now let us look at the random variables $X$ and $Y$ using polar coordinates.

$X = r\cos(\theta)$, $Y = r \sin(\theta)$. note that $X^2 +Y^2= r^2$. If $\theta$ is uniform on $(0,2 \pi)$ and $r$ has a Rayleigh distribution $X$ and $Y$ will be independent normals each with $0$ mean and common variance. The converse is also true. The proof of the converse is what I think the OP wants as the answer to the second part of the question.

Here is a sketch of the proof. Without loss of generality, we can assume that $X$ is distributed $N(0,1)$ and $Y$ is distributed $N(0,1)$ and independent of each other.

Then the joint density $f(x,y) = (1/2 \pi)\exp[(-[x^2 +y^2])/2]$. Use the transformation to polar coordinates to get $g(r, \theta)$. Since $x = r \sin(\theta)$ and $y = r \cos(\theta)$. So $r= \sqrt{x^2 + y^2}$ and $\theta = \arctan(x/y)$. Compute the Jacobian of the transformation and make the appropriate substitution into $f(x,y)$. as a result $g(r, \theta)$ will be $r \exp[(-r^2)/(2 \pi)]$ for $r\geq0$ and $0\leq\theta\leq 2 \pi$. This shows that $r$ and theta are independent with $r$ having a Rayleigh distribution and theta has the constant density $1/(2 \pi)$.