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Describe in detail a birthanddeath processes. Include in your discussion general examples of particular industries that might employ birthanddeath processes and why? Finally, present a

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Describe in detail a birthanddeath processes. Include in your discussion general examples of particular industries that might employ birthanddeath processes and why? Finally, present a specific example showing in detail where birthanddeath processes can be employed and show a stepwise solution using this theory. 10. Consider a linear growth model with immigration where the birth and death rates are respec- 1'.i'I.re1:1..r given by A\" = n}; +9 and an = up. where A. .m" are positive parameters. Under which eondition[s) this birth and death process is positive recurrent. Justifyr your answer. 4. Consider the Markov chain X" = {X,} with state space S = {0, 1, 2, ...} and transition probabilities 1 ifj=i-1 Puj = 10 otherwise , for i 2 1 and Poo = 0, Poj = for j > 1. (a) Is this Markov chain irreducible? Determine the period for every state. (b) Is the Markov chain recurrent or transient? Explain. (c) Is the Markov chain positive recurrent? If so, compute the sta- tionary probability distribution. (d) For each state i, what is the expected number of steps to return to state i if the Markov chain X starts at state i? 5. Consider a Markov chain X = {X} with state space S = {0, 1, 2, ...} and transition probability matrix 0 1 0 0 P 0 0 P = O p 0 q 0 0 . . . 0 0 P 0 4 0 Here p > 0, q > 0 and p+q =1. Determine when the chain is positive recurrent and compute its stationary distribution.4. (6 points) Consider the following graph of a probability density function: 1.0 (x) 6 0'0 0.0 0.5 From which of the following distributions does the probability density function graphed above arise? A. the beta distribution with parameters o = 3 and 3 = 3 B. the beta distribution with parameters o = 5 and 3 = 5 C. the normal distribution with parameters / = 0.3 and o = 0.5 D. the beta distribution with parameters o = 2 and 3 = 3

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