Timeline for Variance of normalized random vector

11 events

when toggle format	what		by	license	comment
Jul 10 at 21:17	audit	Suggested edits
Jul 10 at 21:19
Jun 26 at 7:20	vote	accept	fennel
Jun 25 at 21:06	answer	added	Michael		timeline score: 1
Jun 25 at 19:48	comment	added	fennel		Thanks for your answer! I got the answer as you suggested. Using $E[Y_1 + \cdots + Y_m] = 0$ and $E[Y^2_1 + \cdots + Y^2_m] = 1$ with the fact that the $Y_i$ are identically distributed allows to derive $E[Y_i] =0$ and $V[Y_i]=E[Y_i^2]=1/n$
Jun 25 at 17:36	comment	added	Michael		Strictly speaking, we should define $Y_i=0$ in the case the denominator is zero (which happens with prob 0 if $n\geq 2$ and $X_i$ has a continuous distribution).
Jun 25 at 17:29	comment	added	Michael		In that case, you can get a simple answer for mean and variance. You can use symmetry to compute $E[Y_i]$ and $E[Y_i^2]$, where I define $$Y_i=\frac{X_i-\overline{X}}{\|\|X-e_n\overline{X}\|\|_2}$$ Notice that $\sum_{i=1}^nY_i$ and $\sum_{i=1}^nY_i^2$ are nice.
Jun 25 at 15:27	comment	added	fennel		@Michael Sorry for the ambiguity. $e_n$ is the $n \times 1$ vector whose entries are all 1.
Jun 25 at 15:26	history	edited	fennel	CC BY-SA 4.0	added 27 characters in body
Jun 25 at 15:17	comment	added	Michael		What is the $e_n$?
Jun 25 at 14:49	history	edited	fennel	CC BY-SA 4.0	deleted 11 characters in body
Jun 25 at 14:36	history	asked	fennel	CC BY-SA 4.0