normalized standard gaussian random vectors have the orthogonally invariant distribution on the unit sphere

[[concept]]

Topics

const fieldName = "theme"; // Your field with links
const oldPrefix = "Thoughts/01 Themes/";
const newPrefix = "Digital Garden/Topics/";
 
const relatedLinks = dv.current()[fieldName];
 
if (Array.isArray(relatedLinks)) {
    // Map over the links, replace the path, and output only clickable links
    dv.el("span",
        relatedLinks
            .map(link => {
                if (link && link.path) {
                    let newPath = link.path.startsWith(oldPrefix)
                        ? link.path.replace(oldPrefix, newPrefix)
                        : link.path;
                    return dv.fileLink(newPath);
                }
            })
            .filter(Boolean).join(", ") 
	// Remove any undefined/null items
    );
} else {
    dv.el(dv.current().theme);
}
 
 

Theorem

If $x\sim \mathcal{N}(0,I_d)$, then $\text{Law}\left(\frac{x}{||x||}\right)=\text{Unif}({\mathbb{S}}^{d-1})$

Proof

The result follows immediately from standard gaussian random vectors are orthogonally invariant and the independence proposition for orthogonally invariant distribution on the unit sphere.

References

rmt

What is the important takeaway from the statement “normalized standard gaussian random vectors have the orthogonally invariant distribution on the unit sphere” ? -?- The direction of a gaussian random vector is uniformly distributed

References

See Also

Mentions

Mentions

const modules = await cJS()
 
const COLUMNS = [  
	{ id: "Name", value: page => page.$link },  
	{ id: "Last Modified", value: page => modules.dateTime.getLastMod(page) },
];  
  
return function View() {  
	const current = dc.useCurrentFile();
// Selecting `#game` pages, for example. 
	let queryString = `@page and linksto(${current.$link})`;
	let pages = dc.useQuery(queryString);
	
	// check types
	pages = pages.filter( (p) => !modules.typeCheck.checkAll(p, current) ).sort()
	
	
	return <dc.Table columns={COLUMNS} rows={pages} paging={20}/>;  
}  

const { dateTime } = await cJS()
 
return function View() {
	const file = dc.useCurrentFile();
	return <p class="dv-modified">Created {dateTime.getCreated(file)}     ֍     Last Modified {dateTime.getLastMod(file)}</p>
}