Ehrenfest model of diffusion. We have a total of n balls, some of them black, some white. At each time step, we either do nothing, which happens with probability f, where 0 < f < 1, or we select a ball at random, so that each ball has probability (1 - f ) / n > 0 of being selected. In the latter case, we change the color of the selected ball (if white it becomes black, and vice versa), and the process is repeated indefinitely. What is the steady-state distribution of the number of white balls?

Lecture 11 Nicole Rubenstein October 10, 2017 Contents 1 Continuous random variables 1 1.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1. . . . . . . . 2 Properties 2 3 Probability density function 2 3.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . . . . . . . . 3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . . . . . . . 4 Quantile