A frog is in a pond with 5 water lilies numbered from 1 to 5. With exponential rate 1, the frog leaves its current water lily, chooses a new one uniformly among the four others and jumps to it. We assume the frog starts from lily 1. Let X(t) be the number of the lily where the frog is at time t. a. Admitting that X(t) is a continuous-time Markov chain, give its parameters (i.e. the v; and pij of the course). No proof is required. b. Let pij(t) = P(X(t) = j|X(0) = 1). Explain why P12(t) = P13(t) = P14(t) = P15(t) (no computations required). c. Write the backward Chapman-Kolmogorov equation, and prove that d. 15 Pu(t)=-Pu(t). 4 Solve this equation to compute pii(t).

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