31.1. Introduction http://www.ck12.org
When the number of steps becomes large, the distribution begins to look like a bell curve; here is the plot for
N=100:
Problems
- What is the difference, in terms of end probability distributions, between random walks with even and odd
numbers of steps? - Solve for the probability mass function of end locations for a four-step random walk analogously to the three-
step example above (illustrating it also). Then, graph this probability mass function. - Our proof for the general case can be called ’right-biased’ in two ways. This question settles both:
a. We found the probability of beingDto the right of the origin, but the probability distributions were
graphed as symmetrical. First, explain why this must be true in terms of possible outcomes and end
locations. Then, show that the formula forP(D)can be used to find probabilities to the left also, that is,
prove thatP(D) =P(−D)using the formula above and the definition of factorials.
b. We also found the number of outcomes that lead to an end displacement ofDin terms of steps taken to
the right. Use the result from the previous part to show that using 1/ 2 (N−D)— the number of steps to
the left corresponding to a final distance ofDsteps to the right — in the derivation of the general result
would not have changed it.
a. Derive the probability mass function for a biased random walk (that is, steps in one direction are more
likely than in the other, or the coin has a higher probability of landing heads (Pthan tails (1−P). (Hint
1: Does the number of outcomes for a specific end location change?) (Hint 2: ALL the outcomes will no
longer be equally likely, but what about outcomes that lead to specific end locations?) (Hint 2.5: every
outcome that leads to a specific location HAS to have the same number of steps left and right).
b. Graph a few of these distributions.