Timeline for Expected number of trials to cover certain probability mass for a probability density function?

6 events

when toggle format	what		by	license	comment
Mar 8, 2012 at 22:02	vote	accept	nil
Mar 8, 2012 at 17:45	history	edited	Anthony Quas	CC BY-SA 3.0	added 14 characters in body
Mar 8, 2012 at 17:45	comment	added	Anthony Quas		Same argument. Just split up $\mathbb R^d$ into cubes of diameter less than $\epsilon$. There are still countably many cubes so there's a way to number them 1,2,3,4,\ldots.
Mar 8, 2012 at 16:11	comment	added	nil		Sorry for the confusion, but in 2), I wanted to say $\mathbb{R}^d$, a d-dimensional random variable, not $\mathbb{R}^n$.
Mar 8, 2012 at 15:42	comment	added	nil		Thank you very much for the reply. I have two more concerns about generalising this proof however. 1) If we make $\epsilon\to 0$, (i.e. change the $\sum_k$ to integral), does this still hold due to monotone convergence theorem? (I didn't get to learn real analysis quite well as an engineering student.) 2) If we take $X$ to be a multivariate random variable on $\mathbb{R}^n$, we need then generalise the intervals to $\epsilon$ balls. Does the proof need some kind of covering theorem (I'm guessing) to support it?
Mar 8, 2012 at 5:40	history	answered	Anthony Quas	CC BY-SA 3.0