# Let t exponential ie have an 77 78 chapter 2 poisson

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Unformatted text preview: or m 1 p(m, m for m 2 for m 2 1) = 1/2 p(m, m) = 1/(2m + 2) and p(1, 1) = 1 p(1, 2) = 3/4. Show that there is no stationary distribution. 76 CHAPTER 1. MARKOV CHAINS 1.74. Consider the aging chain on {0, 1, 2, . . .} in which for any n 0 the individual gets one day older from n to n + 1 with probability pn but dies and returns to age 0 with probability 1 pn . Find conditions that guarantee that (a) 0 is recurrent, (b) positive recurrent. (c) Find the stationary distribution. 1.75. The opposite of the aging chain is the renewal chain with state space {0, 1, 2, . . .} in which p(i, i 1) = 1 when i &gt; 0. The only nontrivial part of the transition probability is p(0, i) = pi . Show that this chain is always recurrent P but is positive recurrent if and only if n npn &lt; 1. 1.76. Consider a branching process as deﬁned in Example 7.2, in which each family has exactly three children, but invert Galton and Watson’s original motivation and ignore male children. In this model a mother will have an average of 1.5 daughters. Compute the probability that a given woman’s descendent...
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