1

we know that $(X_k)$ i.i.d, $N(0,1)$, then what is the distribution of the vector $(\frac{X_1}{\sqrt{X_1^2+X_2^2+\cdots+X_n^2}},\frac{X_2}{\sqrt{X_1^2+X_2^2+\cdots+X_n^2}},\cdots,\frac{X_n}{\sqrt{X_1^2+X_2^2+\cdots+X_n^2}})$

I Know that the vector sums up to 1, but it seems not helpful. I have no idea what to do next.

user284873
  • 496
  • Are you looking for the distribution of each component, or the distribution of the norm? – Paul Oct 29 '15 at 17:38
  • not the norm nor each component, but the joint distribution of all the component. – user284873 Oct 29 '15 at 17:41
  • By the invariance of the normal vector $(X_k)$ by rotations, this is uniformly distributed on the unit sphere. – Did Oct 29 '15 at 18:33

0 Answers0