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Probability that a von Mises - Fisher vector has a larger angle than a uniform distributed vector
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My calculus is a bit rusty and I need to calculate the following.
I have two independent random vectors $X,Y$.
$X$ is a unit vector, von Mises-Fisher distributed with direction $e_1$ on the $n-1$ dimensional sphere in $mathbb{R}^n$, i.e., $f(x ;mu,kappa) = Cexp(kappa cdot mu^tx)$ with $mu=(1,0,...,0)$ ,$kappa geq0$ and $C$ is a constant dependent on $kappa,n$.
$Y$ is a unit vector, uniformly distributed on the $n-1$ dimensional sphere in $mathbb{R}^n$.
I would like to calculate the probability that the angle $X$ creates with $e_1$ is greater than the angle $Y$ creates with $e_1$ as a function of $kappa$. For example, if $kappa = 0$ then this probability is $frac12$.
How do I approach this?
calculus multivariable-calculus probability-distributions
asked Jan 12 at 13:30
catch22catch22
1,3361122
$endgroup$
add a comment |
$begingroup$
My calculus is a bit rusty and I need to calculate the following.
I have two independent random vectors $X,Y$.
$X$ is a unit vector, von Mises-Fisher distributed with direction $e_1$ on the $n-1$ dimensional sphere in $mathbb{R}^n$, i.e., $f(x ;mu,kappa) = Cexp(kappa cdot mu^tx)$ with $mu=(1,0,...,0)$ ,$kappa geq0$ and $C$ is a constant dependent on $kappa,n$.
$Y$ is a unit vector, uniformly distributed on the $n-1$ dimensional sphere in $mathbb{R}^n$.
I would like to calculate the probability that the angle $X$ creates with $e_1$ is greater than the angle $Y$ creates with $e_1$ as a function of $kappa$. For example, if $kappa = 0$ then this probability is $frac12$.
How do I approach this?
calculus multivariable-calculus probability-distributions
asked Jan 12 at 13:30
catch22catch22
1,3361122
$endgroup$
add a comment |
$begingroup$
My calculus is a bit rusty and I need to calculate the following.
I have two independent random vectors $X,Y$.
$X$ is a unit vector, von Mises-Fisher distributed with direction $e_1$ on the $n-1$ dimensional sphere in $mathbb{R}^n$, i.e., $f(x ;mu,kappa) = Cexp(kappa cdot mu^tx)$ with $mu=(1,0,...,0)$ ,$kappa geq0$ and $C$ is a constant dependent on $kappa,n$.
$Y$ is a unit vector, uniformly distributed on the $n-1$ dimensional sphere in $mathbb{R}^n$.
I would like to calculate the probability that the angle $X$ creates with $e_1$ is greater than the angle $Y$ creates with $e_1$ as a function of $kappa$. For example, if $kappa = 0$ then this probability is $frac12$.
How do I approach this?
calculus multivariable-calculus probability-distributions
asked Jan 12 at 13:30
catch22catch22
1,3361122
$endgroup$
My calculus is a bit rusty and I need to calculate the following.
I have two independent random vectors $X,Y$.
$X$ is a unit vector, von Mises-Fisher distributed with direction $e_1$ on the $n-1$ dimensional sphere in $mathbb{R}^n$, i.e., $f(x ;mu,kappa) = Cexp(kappa cdot mu^tx)$ with $mu=(1,0,...,0)$ ,$kappa geq0$ and $C$ is a constant dependent on $kappa,n$.
$Y$ is a unit vector, uniformly distributed on the $n-1$ dimensional sphere in $mathbb{R}^n$.
I would like to calculate the probability that the angle $X$ creates with $e_1$ is greater than the angle $Y$ creates with $e_1$ as a function of $kappa$. For example, if $kappa = 0$ then this probability is $frac12$.
How do I approach this?
calculus multivariable-calculus probability-distributions
calculus multivariable-calculus probability-distributions
asked Jan 12 at 13:30
catch22catch22
1,3361122
asked Jan 12 at 13:30
catch22catch22
1,3361122
edited Jan 12 at 14:34
catch22
asked Jan 12 at 13:30
catch22catch22
1,3361122
asked Jan 12 at 13:30
catch22catch22
1,3361122
asked Jan 12 at 13:30
catch22catch22
1,3361122
1,3361122
add a comment |
add a comment |
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