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SaintsTheMeta
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This is a tiny part of a presentation I am giving Friday, any and all help is appreciated.
Suppose we have a circle centered on O. We are looking for the distribution of the points generated by the following method:
We choose a random radius of the circle, and then choose points along that radius with a uniform distribution. i.e. for any arbitrary radius of the circle, we will have a uniform distribution of points f(r)=\frac{1}{R}.
My question is how do we extrapolate this to the area of a circle, to account for all "random radiuses?"
ET Jaynes gives a solution: f(r)=\frac{1}{2πRr}
here: http://www.google.com/url?sa=t&rct=...OQeRMbfgQ&sig2=goMBcl6Cr-jKjCMYz11G_g&cad=rja
page 5, last sentence before the section 4 header.
f(r)=\frac{1}{2πRr}
This solution seems logical. But I need to be able to give a FORMAL proof of where this comes from. I just CAN'T seem to understand exactly where this equation comes from.
My thinking is it is a uniform distribution over circumference f(r)=\frac{1}{2\pi r} multiplied with the uniform distribution of each radius \frac{1}{R}
Or, we could integrate \frac{1}{R} over angles 0 to 2pi which would give what we want, but Whyyyy
Can anyone help me PRECISELY understand this? This is the kind of thing that is almost always "hand waved" in physics, and while it seems logical, formalism is difficult for me after my studies focused on physics.
Thank you!
edit: My best explanation at this point would be a series of "rings" with circumference 2∏r, then multiplied with the uniform distribution 1/R. But still, this is not logically sound, argh... Why would it be 1 over 2piR?? And what could allow you to just multiply these expressions together??
Homework Statement
Suppose we have a circle centered on O. We are looking for the distribution of the points generated by the following method:
We choose a random radius of the circle, and then choose points along that radius with a uniform distribution. i.e. for any arbitrary radius of the circle, we will have a uniform distribution of points f(r)=\frac{1}{R}.
My question is how do we extrapolate this to the area of a circle, to account for all "random radiuses?"
ET Jaynes gives a solution: f(r)=\frac{1}{2πRr}
here: http://www.google.com/url?sa=t&rct=...OQeRMbfgQ&sig2=goMBcl6Cr-jKjCMYz11G_g&cad=rja
page 5, last sentence before the section 4 header.
Homework Equations
f(r)=\frac{1}{2πRr}
The Attempt at a Solution
This solution seems logical. But I need to be able to give a FORMAL proof of where this comes from. I just CAN'T seem to understand exactly where this equation comes from.
My thinking is it is a uniform distribution over circumference f(r)=\frac{1}{2\pi r} multiplied with the uniform distribution of each radius \frac{1}{R}
Or, we could integrate \frac{1}{R} over angles 0 to 2pi which would give what we want, but Whyyyy
Can anyone help me PRECISELY understand this? This is the kind of thing that is almost always "hand waved" in physics, and while it seems logical, formalism is difficult for me after my studies focused on physics.
Thank you!
edit: My best explanation at this point would be a series of "rings" with circumference 2∏r, then multiplied with the uniform distribution 1/R. But still, this is not logically sound, argh... Why would it be 1 over 2piR?? And what could allow you to just multiply these expressions together??
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