- #1
coolnessitself
- 35
- 0
Hi all,
I have a question about the actual value associated with the probability p(r) where p(r) is infinite for r=0.
I realize that this p(r) can only be a distribution and only exist under an integral, and can't represent a pdf. My p(r) is a radially symmetric laplace distribution in 2d, centered at the origin:
[tex]
p(r) = \frac{1}{2\pi} K_0\left(r\right)
[/tex]
where [tex]K_0(\cdot)[/tex] is a modified Bessel function of the first kind (recall that [tex]K_0(0)=\infty[/tex]). This exact distribution isn't really my question, it's just one with nonzero probability around r=0 and infinite 'probability' at 0.
All moments are defined, and
[tex]
2\pi \int\limits_0^{\infty} p(r) rdr = 1.
[/tex]
But if I sample from this distribution, will I always get r=0, since p(0) is infinitely more likely than any other r? I see the relation to a dirac delta, but in that case [tex]p(r)=0[/tex] for all r other than r=0. Here, [tex]p(r\ne 0)>0[/tex]. Does that make any difference?
In the end, I'm trying to assign a probability to any r. This would be easy if all p(r)<=1, but I don't know what to do in this case. Any question I can think of asking hits a roadblock since the pdf doesn't exist, but I guess I'm simply having trouble interpreting what that means if some p(r) are nonzero and finite.
I have a question about the actual value associated with the probability p(r) where p(r) is infinite for r=0.
I realize that this p(r) can only be a distribution and only exist under an integral, and can't represent a pdf. My p(r) is a radially symmetric laplace distribution in 2d, centered at the origin:
[tex]
p(r) = \frac{1}{2\pi} K_0\left(r\right)
[/tex]
where [tex]K_0(\cdot)[/tex] is a modified Bessel function of the first kind (recall that [tex]K_0(0)=\infty[/tex]). This exact distribution isn't really my question, it's just one with nonzero probability around r=0 and infinite 'probability' at 0.
All moments are defined, and
[tex]
2\pi \int\limits_0^{\infty} p(r) rdr = 1.
[/tex]
But if I sample from this distribution, will I always get r=0, since p(0) is infinitely more likely than any other r? I see the relation to a dirac delta, but in that case [tex]p(r)=0[/tex] for all r other than r=0. Here, [tex]p(r\ne 0)>0[/tex]. Does that make any difference?
In the end, I'm trying to assign a probability to any r. This would be easy if all p(r)<=1, but I don't know what to do in this case. Any question I can think of asking hits a roadblock since the pdf doesn't exist, but I guess I'm simply having trouble interpreting what that means if some p(r) are nonzero and finite.