Mixing
Probability that random hyperplane separates two random points
I don't have a good intuition for the following problem:
Given three random (unit) vectors $a, b, n in mathbb{R}^d$, chosen u.a.r. as points on a unit sphere, with $n$ being the normal vector of a hyperplane $h_n$.
As the dimension $d$ increased, does the probability of $h_n$ separating $a$ from $b$ increase, decrease or stays the same?
geometry random-variables
asked Nov 22 '18 at 0:30
stefanstefan
4201418
add a comment |
I don't have a good intuition for the following problem:
Given three random (unit) vectors $a, b, n in mathbb{R}^d$, chosen u.a.r. as points on a unit sphere, with $n$ being the normal vector of a hyperplane $h_n$.
As the dimension $d$ increased, does the probability of $h_n$ separating $a$ from $b$ increase, decrease or stays the same?
geometry random-variables
asked Nov 22 '18 at 0:30
stefanstefan
4201418
add a comment |
I don't have a good intuition for the following problem:
Given three random (unit) vectors $a, b, n in mathbb{R}^d$, chosen u.a.r. as points on a unit sphere, with $n$ being the normal vector of a hyperplane $h_n$.
As the dimension $d$ increased, does the probability of $h_n$ separating $a$ from $b$ increase, decrease or stays the same?
geometry random-variables
asked Nov 22 '18 at 0:30
stefanstefan
4201418
I don't have a good intuition for the following problem:
Given three random (unit) vectors $a, b, n in mathbb{R}^d$, chosen u.a.r. as points on a unit sphere, with $n$ being the normal vector of a hyperplane $h_n$.
As the dimension $d$ increased, does the probability of $h_n$ separating $a$ from $b$ increase, decrease or stays the same?
geometry random-variables
geometry random-variables
asked Nov 22 '18 at 0:30
stefanstefan
4201418
asked Nov 22 '18 at 0:30
stefanstefan
4201418
asked Nov 22 '18 at 0:30
stefanstefan
4201418
asked Nov 22 '18 at 0:30
stefanstefan
4201418
asked Nov 22 '18 at 0:30
stefanstefan
4201418
4201418
add a comment |
add a comment |
1 Answer
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It stays the same. Choose first the hyperplane $H$ randomly. The probability of $ain H$ or $bin H$ is $0$. The hyperplane separates the sphere in two hemisphers so in particular the probability of $a$ being in one of them is $1/2$ and the same is true for $b$. So essentially we have two independent bernoulli random variables.
answered Nov 22 '18 at 0:47
Dante GrevinoDante Grevino
94319
add a comment |
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1 Answer
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1 Answer
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It stays the same. Choose first the hyperplane $H$ randomly. The probability of $ain H$ or $bin H$ is $0$. The hyperplane separates the sphere in two hemisphers so in particular the probability of $a$ being in one of them is $1/2$ and the same is true for $b$. So essentially we have two independent bernoulli random variables.
answered Nov 22 '18 at 0:47
Dante GrevinoDante Grevino
94319
add a comment |
It stays the same. Choose first the hyperplane $H$ randomly. The probability of $ain H$ or $bin H$ is $0$. The hyperplane separates the sphere in two hemisphers so in particular the probability of $a$ being in one of them is $1/2$ and the same is true for $b$. So essentially we have two independent bernoulli random variables.
answered Nov 22 '18 at 0:47
Dante GrevinoDante Grevino
94319
add a comment |
It stays the same. Choose first the hyperplane $H$ randomly. The probability of $ain H$ or $bin H$ is $0$. The hyperplane separates the sphere in two hemisphers so in particular the probability of $a$ being in one of them is $1/2$ and the same is true for $b$. So essentially we have two independent bernoulli random variables.
answered Nov 22 '18 at 0:47
Dante GrevinoDante Grevino
94319
It stays the same. Choose first the hyperplane $H$ randomly. The probability of $ain H$ or $bin H$ is $0$. The hyperplane separates the sphere in two hemisphers so in particular the probability of $a$ being in one of them is $1/2$ and the same is true for $b$. So essentially we have two independent bernoulli random variables.
answered Nov 22 '18 at 0:47
Dante GrevinoDante Grevino
94319
edited Nov 25 '18 at 16:42
answered Nov 22 '18 at 0:47
Dante GrevinoDante Grevino
94319
answered Nov 22 '18 at 0:47
Dante GrevinoDante Grevino
94319
answered Nov 22 '18 at 0:47
Dante GrevinoDante Grevino
94319
94319
add a comment |
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