(a) Mn+1=1+(1-2/M)μn. (b) Use (a) to prove that M Pa = 7/7+ Mn 2 2 M (M7²)" (EIXO) - 4/1) 2

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54. Consider the Ehrenfest urn model in which M molecules are distributed be-
tween 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
Mn = E[X₂]. Show that
(a) Mn+1=1+ (1 – 2/Μ)μη·
(b) Use (a) to prove that
Mn =
n
M
4/1 + (M = ²)" (E[Xo] - M1)
쓸)
2
Transcribed Image Text:54. Consider the Ehrenfest urn model in which M molecules are distributed be- tween 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 Mn = E[X₂]. Show that (a) Mn+1=1+ (1 – 2/Μ)μη· (b) Use (a) to prove that Mn = n M 4/1 + (M = ²)" (E[Xo] - M1) 쓸) 2
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