Binomial random walk

After N steps, the net displacement of the particle (along the x-axis) is given by 

The probability function of the stochastic variable Yjv is then expressed by 

We can find a differential equation for the characteristic function in the limit of both the step size s and the time between steps, At, becoming infinitely small. Let 

Application of the inverse Fourier transformation to Eq. (3.D.14) gives the probability distribution function for the particle to accumulate a net displacement y  

and Fischer, E. W., in Polymer Motion in Dense Systems (eds. D. Richter and T. Springer), Springer-Verlag, Berlin (1987). 

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The random-walk model of diffusion can also be applied to derive the shape of the bell-shaped concentration profile characteristic of bulk diffusion. As in the previous section, a planar layer of N tracer atoms is the starting point. Each atom diffuses from the interface by a random walk of n steps in a direction perpendicular to the interface. As mentioned (see footnote 5) the statistics are well known and described by the binomial distribution (Fig. S5.5a-S5.5c). At large values of N, this discrete distribution can be approximated by a continuous function, the Gaussian distribution curve7 with a form ... 

Figure S5.5 Random-walk statistics Each plot shows the number of atoms, N, reaching a distance d in a random walk, for walks (a) 100 atoms and 200 steps, (b) 500 atoms and 200 steps, and (c) 10,000 atoms and 400 steps. The curve approximates to the binomial distribution as the number of atoms and steps increases.

For particles executing a random walk, like albumin molecules in buffered water, the calculations above suggest that individual steps in the random walk occur very quickly, over a short time interval. As a consequence, during a typical observation time, each particle takes many steps on the axis of Figure 3.2. The probability that a random walking particle took a total of k steps to the right after a sequence of n steps in the random walk is provided by the binomial distribution ... 

By using the binomial distribution, it is possible to determine p(x), the probability of finding a particle at a position between x and x + dx, as a function of time after a large number of individual steps in the random walk have occurred [2] ... 

Table 3.1 Characterization of random walks with binomial distribution...

The binomial distribution, Equation 3-13, was used to calculate the probability of finding a random walker at position x after 4 steps in an unbiased random walk. The position on a coordinate axis, x, was determined from the number of steps to the right, k, and number of steps to the left, n — k ... 

In the first step, we determine the interest rate path in which we create a risk-neutral recombining lattice with the evolution of the 6-month interest rate. Therefore, the nodes of the binomial tree are for each 6-month interval, and the probability of an upward and downward movement is equal. The analysis of the interest rate evolution has a great relevance in callable bond pricing. We assume that the interest rate follows the path shown in Figure 11.4. In this example, we assume for simplicity a 2-year interest rate. We suppose that the interest rate starts at time tg and can go up and down following the geometric random walk for each period. The interest rate rg at time tg changes due to two main variables ... 

The following section illustrates the Taylor series method, and also introduces an important model in statistical thermodynamics the random walk (in two dimensions) or random flight (in three dimensions). In this example, we find that the Gaussian distribution function is a good approximation to the binomial distribution function (see page 15) when the number of events is large. 

Physical theories often require mathematical approximations. When functions are expressed as polynomial series, approximations can be systematically improved by keeping terms of increasingly higher order. One of the most important expansions is the Taylor series, an expression of a function in terms of its derivatives. These methods show that a Gaussian distribution function is a second-order approximation to a binomial distribution near its peak. We will hnd this useful for random walks, which are used to interpret diffusion, thermal conduction, and polymer conformations. In the next chapter we develop additional mathematical tools. 

We have seen that the binomial distribution of a random walk reduces to the Gaussian distribution as n c . We now make this result more general. 

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