Problem 1 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. 1. Admitting that X (t) is a continuous-time Markov chain, give its parameters (i.e. the vi and pig- of the course). No proof is required. 2. Let p1j(t) = lP'(X(t) = j|X(0) = 1). Explain why p12(t) = p13(t) = p14(t) = p15(t) (no compu- tations required). 3. Write the backward ChapmanKolmogorov equation, and prove that 1 5 Fina) = Z 11911\")- 4. Solve this equation to compute p11(t)