I would've thought that this random sampling method intuitively makes sense at a first glance, so it was surprising that it turns out that it doesn't actually uniformly take samples.
Bananya
Yeah, it's trickier than I expected. I like the clear explanation here.
sponge
I'm still not sure I understand why sampling is not uniform in area. Why are points closer to the origin more likely to be chosen?
dranzer
Yup can we have more clarification for that. Connecting it to the notion of area doesn't come out of the mathematics. I was thinking along the lines of the probability of a point to be picked in a circle. Since all the points are made up of unique r and theta they should all have the same probability. It's not immediately clear to me why closer points have higher probability. Is it because of the CDF ?
marshmallow
I think it's because in a uniform distribution, you want each point in a given bound to all be equally likely, and in the case of a circle, the area is given by pi * r^2, so it's proportional to r^2. But, we're picking points inside the circle with a probability proportional to r, not r^2, so we're more likely to pick points closer to the center than points farther from the center.
keenan
@marshmallow has it right. You could think of this in a more discrete way: suppose you break up the disk into $n$ concentric rings, and pick and equal number of points in each ring. The points on the inner rings will be more densely packed than in the outer rings. That's all.
I would've thought that this random sampling method intuitively makes sense at a first glance, so it was surprising that it turns out that it doesn't actually uniformly take samples.
Yeah, it's trickier than I expected. I like the clear explanation here.
I'm still not sure I understand why sampling is not uniform in area. Why are points closer to the origin more likely to be chosen?
Yup can we have more clarification for that. Connecting it to the notion of area doesn't come out of the mathematics. I was thinking along the lines of the probability of a point to be picked in a circle. Since all the points are made up of unique r and theta they should all have the same probability. It's not immediately clear to me why closer points have higher probability. Is it because of the CDF ?
I think it's because in a uniform distribution, you want each point in a given bound to all be equally likely, and in the case of a circle, the area is given by pi * r^2, so it's proportional to r^2. But, we're picking points inside the circle with a probability proportional to r, not r^2, so we're more likely to pick points closer to the center than points farther from the center.
@marshmallow has it right. You could think of this in a more discrete way: suppose you break up the disk into $n$ concentric rings, and pick and equal number of points in each ring. The points on the inner rings will be more densely packed than in the outer rings. That's all.