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1. Biased Random Walker in One Dimension: Consider a particle that moves in one dimension, taking steps always in the same direction, but with step lengths s equally likely to be anywhere in the range l - b < s < l + b (b < l). After N steps, what is (a) the mean value of the displacement, x = sum_{i=1}^{N} s_i, from the origin? (b) the dispersion (variance) overline{(x - ar{x})^2} in the displacement? (c) Now suppose that the step length has a Gaussian distribution, such that the probability of a step length between s and s + ds is given by p(s)ds = frac{1}{sqrt{2pi}sigma} exp left[ frac{-(s - l)^2}{2sigma^2} ight] ds . After N steps, what are the mean displacement ar{x} from the origin and the dispersion overline{(x - ar{x})^2} ?

          1. Biased Random Walker in One Dimension:
Consider a particle that moves in one dimension, taking steps always in the same direction, but with step lengths s equally likely to be anywhere in the range l - b < s < l + b (b < l). After N steps, what is
(a) the mean value of the displacement, x = sum_{i=1}^{N} s_i, from the origin?
(b) the dispersion (variance) overline{(x - ar{x})^2} in the displacement?
(c) Now suppose that the step length has a Gaussian distribution, such that the probability of a step length between s and s + ds is given by
p(s)ds = frac{1}{sqrt{2pi}sigma} exp left[ frac{-(s - l)^2}{2sigma^2} 
ight] ds .
After N steps, what are the mean displacement ar{x} from the origin and the dispersion overline{(x - ar{x})^2} ?

$1. Biased Random Walker in One Dimension: Consider a particle that moves in one dimension, taking steps always in the same direction, but with step lengths s equally likely to be anywhere in the range l - b < s < l + b (b < l). After N steps, what is (a) the mean value of the displacement, x = sumi=1^N si, from the origin? (b) the dispersion (variance) overline(x - arx)^2 in the displacement? (c) Now suppose that the step length has a Gaussian distribution, such that the probability of a step length between s and s + ds is given by p(s)ds = frac1sqrt2pisigma exp left[ frac-(s - l)^22sigma^2 ight] ds . After N steps, what are the mean displacement arx from the origin and the dispersion overline(x - arx)^2 ?$

Added by Kimberly H.

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