[b] In {one of!) Einstein's seminal work[s], he determined the probability for a Brownian particle to he at a position (x, y) relative to its starting point [0,0]: ?[x.y) = exp [TJxZ + fl] . (2} where D is the \"diffusion constant" (which is a function of temperature], and t is the time. Using that the average value of a quantity x. 3.?) is obtained via (x. 39)) = I dxdyx. y)f(x, 3r} . Where 390:. y) is the probability distribution. compute {x2 + 3:2) {with 590:, 3:) given by Eq. {2). Consider a pollen particle (Le. a small piece of dirt] moving on the surface of a cup of water. Because the pollen particle is bombarded by the water molecules, it exhibits Brownian motion the pollen particle moves randomly on the water's surface. To model this Brownian motion, we will consider a particle exhibiting a random walk ~ the particle starts at the origin; at each step [of the walkji. one of 4 \"moves\" is chosen at random: [1] one step in positive-x [2] one step in negative-x [2] one step in positive-y [4] one step in negative-y