• Use statistical software R for your codes.
• Only basic built-in functions available in R are allowed.
• In each question, show the simulations with relevant graphs.
1. Suppose the interarrival times of a renewal process have the geometric distribution with success probability p. Then the interarrival times are nonnegative integers, and sum of interarrival times is Sn = X1 + ... + Xn. Answer the following.
(a) Simulate sum of interarrival times with p = 0.2.
(b) Simulate Nt and estimate E(Nt) for various values of t. Use p = 0.2.
2. Let there be a discrete time Markov chain with the state space
S = {0,1,...,7}. The one step transition probability is given by
2/3
1/3
0
1/3
0
0
0
0
0
0
0
0
0 1/3 1/3 1/3 0 0 0 0
0 0 1/3 1/3 1/3 0 0 0 P = 0 0 0 1/3 1/3 1/3 0 0
0 0 0 0 1/3 1/3 1/3 0
0 0 0 0 0 1/3 1/3 1/3
0 0 0 0 0 0 2/3 1/3
It is given that when the process starts the MC was in state 0. Answer the following.
(a) Simulate five times a 50 steps Markov chain. Construct a plot comparing time to the states of the process.
(b) What will be P10, P20, P50. What do you observe.
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