

A137272


Number of 1D random walks with 8 steps where the median of the positions is n.


3




OFFSET

4,1


COMMENTS

Consider the 1D random walk starting at position 0, with equal probability of moving one unit to the left or one unit to the right. This allows 2^s different trajectories if we consider a maximum of s steps (s=8 here). For each of the trajectories, compute the median position, which is in the interval [s/2, +s/2].
The sequence shows the count of trajectories with median equal to n (so the sum over all elements of the sequence is again 2^s = 256).
1) Suppose s is even, the convolution of the probability distribution of the minimum and the maximum of a simple random walk up to s/2 is equal to the probability distribution of the median (see Mathematica program and references).
2) The median taken on partial sums of the simple random walk represents the market price in a simulation model wherein a single security among noncooperating and asymmetrically informed traders is traded (see Pfeifer et al. 2009).
3) Transformation T007 gave a match with first differences of A089877 (superseeker).


REFERENCES

W. Feller, An Introduction to Probability Theory and its Applications I. New York: Wiley, 1968.


LINKS



EXAMPLE

The possible different paths (sequences of partial sums) in the case s=2:
{0,1,2}; median=1
{0,1,0}; median=0
{0,1,0}; median=0
{0,1,2}; median=1
Sequence of integers in the case s=2: 1,2,1.
In the current case s=8, we have 6 trajectories with median 4, 10 trajectories with median 3, etc.


MATHEMATICA

(* calculation of distribution of median single random walk *) p[n_, r_] := If[Floor[(n + r)/2]  (n + r)/2 == 0, Binomial[n, (n + r)/2], 0] maximum[n_, r_] := p[n, r] + p[n, r + 1]; (* prob. maximum *) minimum[n_, r_] := p[n, r] + p[n, r + 1]; (* prob. minimum *) median[n_] := ( (* distr. median *) listmin = Table[If[r < (n/2)  r > 0, 0, minimum[n/2, r]], {r, n, n}] (* distr. minimum *); listmax = Table[If[r > n/2  r < 0, 0, maximum[n/2, r]], {r, n, n}] (* distr. maximum *); listmedian = ListConvolve[listmax, listmin, {1, 1}] (* convolution *); listmedian[[3 n/2 + 1 ;; 5 n/2 + 1]]); (* result median *) Table[median[2 n], {n, 1, 7}](* result up to n=14 *)


CROSSREFS



KEYWORD

easy,fini,full,nonn


AUTHOR

Christian Pfeifer (christian.pfeifer(AT)uibk.ac.at), Mar 13 2008, May 03 2010


EXTENSIONS

Variables names normalized, offset set to 4 by R. J. Mathar, Sep 17 2009


STATUS

approved



