For a one dimensional random walk the probability density function W(x) gives the distribution of possible outcomes after N random jumps each of length : W(x)=122e(x)2/22 . where is the average x displacement and 2=2 is the variance. The square root of the variance is the standard deviation . W(x) is the well-known bell-shaped or normal distribution curve.See graph below, right. Suppose the probability of of jumping in the +x direction is p and the probability of jumping in the -x direction is q=1-p.Then after N jumps each of length the average displacement is =N(pq) . After N jumps each of length the variance is 2=4pqN2