Editing Examples of Markov chains (section)

==== A center-biased random walk ====

Consider a [[random walk]] on the number line where, at each step, the position (call it ''x'') may change by +1 (to the right) or −1 (to the left) with probabilities:

: <math>P_{\mathrm{move~left}} = \dfrac{1}{2} + \dfrac{1}{2} \left( \dfrac{x}{c+|x|} \right) </math>

: <math>P_{\mathrm{move~right}} = 1 - P_{\mathrm{move~left}}</math>

(where ''c'' is a constant greater than 0)

For example, if the constant, ''c'', equals 1, the probabilities of a move to the left at positions ''x'' = −2,−1,0,1,2 are given by <math>\dfrac{1}{6},\dfrac{1}{4},\dfrac{1}{2},\dfrac{3}{4},\dfrac{5}{6}</math> respectively. The random walk has a centering effect that weakens as ''c'' increases.

Since the probabilities depend only on the current position (value of ''x'') and not on any prior positions, this biased random walk satisfies the definition of a Markov chain.