Problem 1 (Ehrenfests Diffusion Model). Let 2 be a container separated into the left and the right parts by a barrier in the middle. The container is filled with K particles in total. At each time n = 1, 2, ..., we randomly pick one particle among the K particles and place it into the other part of the container. Let {Xn}nen be the stochastic process where X, is the number of particles in the left part of the container at time n. The state space of the process is therefore X = {0,1, ... , + K}. (i) Explain why {Xn}nen is a Markov process. (ii) Write down the transition probability matrix P for {Xn}nen and analyze its invariant distribution. =