PDF of normalized gaussian random vector
$begingroup$
let $X$ be a Gaussian random vector in $R^n$ such that
$$X sim mathcal{N}(mathbf{0}, mathbf{I_n}),$$
How can I find the PDF of $frac{X}{|X|}$?
probability probability-distributions
asked Jan 10 at 3:26
ShaoyuPeiShaoyuPei
1828
$endgroup$
add a comment |
$begingroup$
let $X$ be a Gaussian random vector in $R^n$ such that
$$X sim mathcal{N}(mathbf{0}, mathbf{I_n}),$$
How can I find the PDF of $frac{X}{|X|}$?
probability probability-distributions
asked Jan 10 at 3:26
ShaoyuPeiShaoyuPei
1828
$endgroup$
add a comment |
$begingroup$
let $X$ be a Gaussian random vector in $R^n$ such that
$$X sim mathcal{N}(mathbf{0}, mathbf{I_n}),$$
How can I find the PDF of $frac{X}{|X|}$?
probability probability-distributions
asked Jan 10 at 3:26
ShaoyuPeiShaoyuPei
1828
$endgroup$
let $X$ be a Gaussian random vector in $R^n$ such that
$$X sim mathcal{N}(mathbf{0}, mathbf{I_n}),$$
How can I find the PDF of $frac{X}{|X|}$?
probability probability-distributions
probability probability-distributions
asked Jan 10 at 3:26
ShaoyuPeiShaoyuPei
1828
asked Jan 10 at 3:26
ShaoyuPeiShaoyuPei
1828
asked Jan 10 at 3:26
ShaoyuPeiShaoyuPei
1828
asked Jan 10 at 3:26
ShaoyuPeiShaoyuPei
1828
asked Jan 10 at 3:26
ShaoyuPeiShaoyuPei
1828
1828
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
By rotation invariance / symmetry, the PDF is the uniform distribution on the unit sphere, so the PDF is the constant value $frac{1}{S_{n-1}}$
where $S_{n-1} = frac{n pi^{n/2}}{Gamma(frac{n}{2}+1)}$ is the surface area of the unit sphere in $mathbb{R}^n$.
answered Jan 10 at 4:12
angryavianangryavian
42.5k23481
$endgroup$
$begingroup$
sorry , Indeed I know this result ,however I can't prove that .and I see some answer on this website but not quite get it .
$endgroup$
– ShaoyuPei
Jan 11 at 1:22
$begingroup$
What can be said when $Xsimmathcal{N}(mu,Sigma)$?
$endgroup$
– nullgeppetto
Feb 20 at 19:43
$begingroup$
@ShaoyuPei math.stackexchange.com/questions/3120506/…
$endgroup$
– angryavian
Feb 21 at 20:34
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
By rotation invariance / symmetry, the PDF is the uniform distribution on the unit sphere, so the PDF is the constant value $frac{1}{S_{n-1}}$
where $S_{n-1} = frac{n pi^{n/2}}{Gamma(frac{n}{2}+1)}$ is the surface area of the unit sphere in $mathbb{R}^n$.
answered Jan 10 at 4:12
angryavianangryavian
42.5k23481
$endgroup$
$begingroup$
sorry , Indeed I know this result ,however I can't prove that .and I see some answer on this website but not quite get it .
$endgroup$
– ShaoyuPei
Jan 11 at 1:22
$begingroup$
What can be said when $Xsimmathcal{N}(mu,Sigma)$?
$endgroup$
– nullgeppetto
Feb 20 at 19:43
$begingroup$
@ShaoyuPei math.stackexchange.com/questions/3120506/…
$endgroup$
– angryavian
Feb 21 at 20:34
add a comment |
$begingroup$
By rotation invariance / symmetry, the PDF is the uniform distribution on the unit sphere, so the PDF is the constant value $frac{1}{S_{n-1}}$
where $S_{n-1} = frac{n pi^{n/2}}{Gamma(frac{n}{2}+1)}$ is the surface area of the unit sphere in $mathbb{R}^n$.
answered Jan 10 at 4:12
angryavianangryavian
42.5k23481
$endgroup$
$begingroup$
sorry , Indeed I know this result ,however I can't prove that .and I see some answer on this website but not quite get it .
$endgroup$
– ShaoyuPei
Jan 11 at 1:22
$begingroup$
What can be said when $Xsimmathcal{N}(mu,Sigma)$?
$endgroup$
– nullgeppetto
Feb 20 at 19:43
$begingroup$
@ShaoyuPei math.stackexchange.com/questions/3120506/…
$endgroup$
– angryavian
Feb 21 at 20:34
add a comment |
$begingroup$
By rotation invariance / symmetry, the PDF is the uniform distribution on the unit sphere, so the PDF is the constant value $frac{1}{S_{n-1}}$
where $S_{n-1} = frac{n pi^{n/2}}{Gamma(frac{n}{2}+1)}$ is the surface area of the unit sphere in $mathbb{R}^n$.
answered Jan 10 at 4:12
angryavianangryavian
42.5k23481
$endgroup$
By rotation invariance / symmetry, the PDF is the uniform distribution on the unit sphere, so the PDF is the constant value $frac{1}{S_{n-1}}$
where $S_{n-1} = frac{n pi^{n/2}}{Gamma(frac{n}{2}+1)}$ is the surface area of the unit sphere in $mathbb{R}^n$.
answered Jan 10 at 4:12
angryavianangryavian
42.5k23481
edited Jan 10 at 4:18
answered Jan 10 at 4:12
angryavianangryavian
42.5k23481
answered Jan 10 at 4:12
angryavianangryavian
42.5k23481
answered Jan 10 at 4:12
angryavianangryavian
42.5k23481
42.5k23481
$begingroup$
sorry , Indeed I know this result ,however I can't prove that .and I see some answer on this website but not quite get it .
$endgroup$
– ShaoyuPei
Jan 11 at 1:22
$begingroup$
What can be said when $Xsimmathcal{N}(mu,Sigma)$?
$endgroup$
– nullgeppetto
Feb 20 at 19:43
$begingroup$
@ShaoyuPei math.stackexchange.com/questions/3120506/…
$endgroup$
– angryavian
Feb 21 at 20:34
add a comment |
$begingroup$
sorry , Indeed I know this result ,however I can't prove that .and I see some answer on this website but not quite get it .
$endgroup$
– ShaoyuPei
Jan 11 at 1:22
$begingroup$
What can be said when $Xsimmathcal{N}(mu,Sigma)$?
$endgroup$
– nullgeppetto
Feb 20 at 19:43
$begingroup$
@ShaoyuPei math.stackexchange.com/questions/3120506/…
$endgroup$
– angryavian
Feb 21 at 20:34
$begingroup$
sorry , Indeed I know this result ,however I can't prove that .and I see some answer on this website but not quite get it .
$endgroup$
– ShaoyuPei
Jan 11 at 1:22
$begingroup$
sorry , Indeed I know this result ,however I can't prove that .and I see some answer on this website but not quite get it .
$endgroup$
– ShaoyuPei
Jan 11 at 1:22
$begingroup$
What can be said when $Xsimmathcal{N}(mu,Sigma)$?
$endgroup$
– nullgeppetto
Feb 20 at 19:43
$begingroup$
What can be said when $Xsimmathcal{N}(mu,Sigma)$?
$endgroup$
– nullgeppetto
Feb 20 at 19:43
$begingroup$
@ShaoyuPei math.stackexchange.com/questions/3120506/…
$endgroup$
– angryavian
Feb 21 at 20:34
$begingroup$
@ShaoyuPei math.stackexchange.com/questions/3120506/…
$endgroup$
– angryavian
Feb 21 at 20:34
add a comment |
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