# continuous vs discrete random walk

For 1D random walk in discrete case the probability of finding walker at position X after N steps(P_N(X)) is binomial distribution(http://mathworld.wolfram.com/RandomWalk1-Dimensional.html), moreover when N+X is odd then probability is 0. Let's take N=7, Sum[P_N(X),{X,-7,7}]=2, but if we take the sum for allowed positions then it's 0: Sum[P_N(X),{X,-7,7,x+=2}]=1. From discrete case we get formula for continuous case and that is Gaussian distribution(P_N(X)=A*Exp[-x^2/2*Nl^2], where l is step length, A stands for normalization factor). The probability of being between X=3, X=6 positions is Sum[P_N(X),{X,3,7,x+=2}], which is not the same if we integrate continuous Gaussian probability from 3 to 7, i.e. Integrate[AExp[-x^2/2*N*l^2],{x,3,7}]. What formula should I use for finding probability of being between some positions in discrete case??? P.S. I'd like in some way to use continuous formula as there is no factorials and also it's important.

• Request: could you clean this question up a bit, it's a little hard to read inline formulas as they are now. – barrycarter Feb 6 '16 at 16:23
• How should I write formulas here? Thank you – Narek Feb 8 '16 at 15:25
• How to write formulas? math.meta.stackexchange.com/questions/5020/… – kjetil b halvorsen Sep 25 '17 at 7:37