# What this kind probability should be called?

I have $m$ continues integer points on a line, randomly uniform select $n$ points from the $m$ point without replacement. Order the points ascendingly. Let the random variable $A_i$ is the position (coordination on the line) of the $i$th point. So, $$P(A_i=k)=\frac{{k-1\choose i-1} {m-k \choose n-i}}{{m \choose n}}$$

How to derive the tail inequality for this probability. The tail probability look something like this:

$$P(|A_i - E(A_i)| > t) < \sigma$$

I want the bound ($\sigma$) to be as tight as possible. The Chebyshev inequality is too loose.

Updated: Some supplement about the question: http://www.math.uah.edu/stat/urn/OrderStatistics.pdf

Your distribution is very close to a hypergeomtric distribution (as noted in an earlier version of the question). In fact, it is related to it via a factor of $i/k$. So tail estimates for it should transfer to tail estimates for your distribution.
• @Yuval: There is at best an analogy between these two settings since the running argument $k$ of the distribution is at the bottom of the binomial coefficient for the hypergeometric distribution and at the top for the distribution Fan is interested in. – Did Feb 15 '11 at 9:12