Consider the master equation for a random walker on a one dimensional lattice of sites with m = 0, plus or minus 1, plus or minus 2, and discrete time t = 0, 1, 2, as discussed in class, p(m)_t + 1 = (1 - p - q)p(m)_t + pp(m - 1)_t + qp(m + 1)_t the particle jumps with probability p(q) to the nearest neighbor site to the right (left), and does nor move with probability 1 - p - q. the probability distribution of the this random walk process, when p + q = 1 has the binomial form p(m, t) =p^n_r q^n_l t! /n_r! n_l! with n_r - n_l! = m and n_r + n_b = t. explain in your own words what this has to do with the binomial expansion of the mathematical form (a + b)^t. also comment on what this has to do with the pascal triangle. the moments of the probability distribution are defined as w_n(t) = ([m - (m)]^n) in class we derived how the first and second moments (n = 1, 2) behave as function of time n starting from a delta function distribution at time tau = 0 or the special binomial case where p + q=1. m_av(t) = (m) = (p - q)t and sigma^2 = ((m - m_av)^2) = 4pqt repeat these derivations here. derive that the third moment of the binomial case p + q = 1 is equal to m_3 = ((m - m_av)^3) = 8[(n^3_r) - 3(n^2_r)(n_r) + 2(n_r)^3] = -8pq(p - q)t with n_r the number of right jumps, such that m = n_r - n_l is the displacement at time t = n_r + n_l.