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fan_bai@yahoo.com
science forum beginner

Joined: 10 Jul 2006
Posts: 1

Posted: Mon Jul 10, 2006 5:03 pm    Post subject: Order Statistic in Random Process/Random Sequence

Dear sci.math,

A sequence of random variables Xi (i=1, 2, ..., N) (Xi>0) follows the
same PDF distribution f(x). Let us assume that another sequence of
random variable Yi= sum_{1}^{i} (Xi).

Suppose we have a straight line. So, we can imagine that Xi is the
space between ith point and i-1th point in the line, and it is obvious
that Yi is the location of ith point (in one-dimension case). Here,
Xi_{1} and Xi_{2} are two instances of the random sequence Xi
following the same probability distributions, thus Yi_{1} and Yi_{2}
are the two instances of random sequences Yi. Actually, Xi_{1} and
Xi_{2} are the spacing distribution of points on two nearby lanes, and
Yi_{1} and Yi_{2} are the locations of points on the two different
lanes.

Imagine that we can sort these two instances of Yi (i.e., Yi_{1} and
Yi_{2} ) into one new random sequence Zi (i=1,2,3, ..., 2N). Obviously,
Zi is an ordered random sequence based on the values of Yi_{1} and
Yi_{2}. Here, the most important metric we aim to capture the PDF
distribution of the distance between Zi and Zi-1(i=1,2,3, ..., 2N) .
We need to derive the PDF distribution of random variable ( Zi - Zi-1
).

As a side note, please note that for any given node Zi (assume that it
is Zi on the 1st lane), Zi-1 could be either the point on 1st lane
following Zi who is immediately larger than Zi, or the point on the 2nd
lane but immediately larger than Zi (it depends who is the closest to
Zi).

Also, what I describe above is only 2-line case, if we have a
three-line case, Xi_{1}, Xi_{2} and Xi_{3}, what is the PDF
distribution of Zi? how about four-line case?

Thanks a lot!

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