How to Maximize Areal Density of Randomly Sized Circles in a Square?

[ManuelCalavera]

Hi,

I'm not sure if this is the right forum to put this question in as it's not a homework problem but it is a math/statistics problem. I'm really not sure how to even start.

So the problem is this:
You have a square with side lengths W and circles that have diameter's that are randomly distributed according to the Gaussian distribution. You can choose the mean of the distribution but the standard dev is set. There is a minimum mean value.

You want to find the mean value of the diameter that will maximize the amount of circles you can fit in the square. The circles can't overlap and they must be whole circles. I'm almost sure it is the minimum value so I guess I just have to prove that.

My background is in engineering/physics so I don't have a first principles math education so I'm really not sure how to even begin the problem.

Thanks for any help (and sorry if this is in the wrong place)

 

[Ray Vickson]

Hi,

I'm not sure if this is the right forum to put this question in as it's not a homework problem but it is a math/statistics problem. I'm really not sure how to even start.

So the problem is this:
You have a square with side lengths W and circles that have diameter's that are randomly distributed according to the Gaussian distribution. You can choose the mean of the distribution but the standard dev is set. There is a minimum mean value.

You want to find the mean value of the diameter that will maximize the amount of circles you can fit in the square. The circles can't overlap and they must be whole circles. I'm almost sure it is the minimum value so I guess I just have to prove that.

My background is in engineering/physics so I don't have a first principles math education so I'm really not sure how to even begin the problem.

Thanks for any help (and sorry if this is in the wrong place)

There are several issues at fault with your formulation, but perhaps they can be corrected, or at least, clarified.
(1) By "quantity of circles", do you mean number of circles, or total area covered by circles?
(2) Are all the circles (a) identical in size, or (b) are they a random sample of independently-chosen circles that are all drawn from the same Gaussian distribution of radius?
(3) If the circles' radii are random, the area covered is a random quantity, so what can you possibly mean by maximizing it? A similar objection occurs if you want to maximize the number of circles that fit in the square.

 

[ManuelCalavera]

There are several issues at fault with your formulation, but perhaps they can be corrected, or at least, clarified.
(1) By "quantity of circles", do you mean number of circles, or total area covered by circles?
(2) Are all the circles (a) identical in size, or (b) are they a random sample of independently-chosen circles that are all drawn from the same Gaussian distribution of radius?
(3) If the circles' radii are random, the area covered is a random quantity, so what can you possibly mean by maximizing it? A similar objection occurs if you want to maximize the number of circles that fit in the square.

Hi,

Thanks for replying. To answer your questions:

1) The areal density, the number of circles per unit area
2) The assumption is that they are taken from a random sample of independently chosen circles that are drawn from the same Gaussian distribution, with a chosen mean and set sigma.
3) You have some discrete amount of circles whose radii is sampled from a gaussian distribution. You're trying to pick a mean value to maximize the areal density. And there is a minimum mean value you can choose.

I think there might still be issues with the formulation of the problem. I think you might be able to get around some of them because I'm almost sure the minimum mean value will always minimize the areal density regardless of how circles are chosen or are placed in the square.

Thanks again