What is the distribution of the unit sphere multiplied by a uniformly distributed scalar?
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We know that any rotationally-invariant distribution can be written as X = RU where R is |X|, U ~ Unif[S^(n-1)], and R is independent of U. In words, we can choose a direction vector using the unit sphere and then choose its length using an independent distribution.
My question is this: Given that R ~ Unif[0,1], how do I find the distribution of X? I can just describe this distribution as the product of R and U, but how do I use it?
probability geometry probability-theory probability-distributions rotations
asked Dec 9 '18 at 20:04
purpleostrichpurpleostrich
53
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$begingroup$
We know that any rotationally-invariant distribution can be written as X = RU where R is |X|, U ~ Unif[S^(n-1)], and R is independent of U. In words, we can choose a direction vector using the unit sphere and then choose its length using an independent distribution.
My question is this: Given that R ~ Unif[0,1], how do I find the distribution of X? I can just describe this distribution as the product of R and U, but how do I use it?
probability geometry probability-theory probability-distributions rotations
asked Dec 9 '18 at 20:04
purpleostrichpurpleostrich
53
$endgroup$
add a comment |
$begingroup$
We know that any rotationally-invariant distribution can be written as X = RU where R is |X|, U ~ Unif[S^(n-1)], and R is independent of U. In words, we can choose a direction vector using the unit sphere and then choose its length using an independent distribution.
My question is this: Given that R ~ Unif[0,1], how do I find the distribution of X? I can just describe this distribution as the product of R and U, but how do I use it?
probability geometry probability-theory probability-distributions rotations
asked Dec 9 '18 at 20:04
purpleostrichpurpleostrich
53
$endgroup$
We know that any rotationally-invariant distribution can be written as X = RU where R is |X|, U ~ Unif[S^(n-1)], and R is independent of U. In words, we can choose a direction vector using the unit sphere and then choose its length using an independent distribution.
My question is this: Given that R ~ Unif[0,1], how do I find the distribution of X? I can just describe this distribution as the product of R and U, but how do I use it?
probability geometry probability-theory probability-distributions rotations
probability geometry probability-theory probability-distributions rotations
asked Dec 9 '18 at 20:04
purpleostrichpurpleostrich
53
asked Dec 9 '18 at 20:04
purpleostrichpurpleostrich
53
asked Dec 9 '18 at 20:04
purpleostrichpurpleostrich
53
asked Dec 9 '18 at 20:04
purpleostrichpurpleostrich
53
asked Dec 9 '18 at 20:04
purpleostrichpurpleostrich
53
53
add a comment |
add a comment |
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