Consider the Ehrenfest urn model in which M molecules are distributed between two urns, and at each time point one of the molecules is chosen at random and is then removed from its urn and placed in the other one. Let Xn denote the number of molecules in urn 1 after the nth switch and let

μn = E[Xn]. Show that

(a) μn+1 = 1+(1 − 2/M)μn.

(b) Use

(a) to prove that

Transcribed Image Text:

n M M-2 n + 2 M E[Xo] - M 2