What is the distribution of the unit sphere multiplied by a uniformly distributed scalar?
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
purpleostrich
53
add a comment |
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
purpleostrich
53
add a comment |
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
purpleostrich
53
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
purpleostrich
53
asked Dec 9 '18 at 20:04
purpleostrich
53
asked Dec 9 '18 at 20:04
purpleostrich
53
asked Dec 9 '18 at 20:04
purpleostrich
53
asked Dec 9 '18 at 20:04
purpleostrich
53
53
add a comment |
add a comment |
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3032908%2fwhat-is-the-distribution-of-the-unit-sphere-multiplied-by-a-uniformly-distribute%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3032908%2fwhat-is-the-distribution-of-the-unit-sphere-multiplied-by-a-uniformly-distribute%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
