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Practice problems for midterm: Stats 217

1. Imagine that you and your friend live in a neighborhood where houses are located on a straight line at integer points, with the house number equal to the integer on which it stands. His house number is 0 while yours is 6. Imagine that he starts running from his house and meets you at your house after an hour. You know from experience that he takes 5 minutes to run from one house to the next. You also know that he is quite whimsical and often runs back and forth between houses and hence he does not necessarily go straight to the destination. After meeting you, he brags about his running skills and tells you that he ran till house 10 before meeting you. Do you believe him? If you do, guess a possible path he took to reach you. If you don’t, explain why. What if he told you he ran till house 8?

2. Establish the following identities:

• Let {Sn }n≥0  denote the simple symmetric random walk on Z with S0   = 0. For any positive integers n, y,

P(Sn  > y) = y   −

• For any integer y > 0,

N≥y:is even  ( y )2 −N  = 1.

3. Consider a gambler who uses two fair coins in a betting game. At every step, he tosses the first coin. If it lands heads, he tosses the second coin, winning $1 if the second coin comes up heads and losing $1 if the second coin comes up tails.  If the first coin shows tails, he neither wins nor loses anything. Find the probability that he will be up by $200 before being down by $100.

4. Using the pmf of τ0  derived in class, show that E(τ0 |S0  = 0) = ∞ . So the SSRW takes, on average, infinite time to return to the origin i.e. its starting point. Does this contradict the fact that the SSRW is recurrent in dimension 1?

5. Consider a branching process starting with one individual and with progeny distribution Bernoulli(p) i.e. 1 offspring with probability p and no offspring with probability 1 − p. Let T be the first time of extinction. Find the distribution of T and E(T).

6.  Suppose that in a branching process, the progeny distribution is Geometric(p).  Compute the extinction probability.

7. For a branching process starting with one individual and with mean progeny size µ ≤ 1, find the expected total population size.  Note that the process gets extinct almost surely so the total population size is finite (unless ξ = 1 almost surely), but is the expectation also finite?

8. Let {Xn }n≥0  denote a discrete time markov chain with state space {0, 1, 2} and transition probability matrix

Compute P(X3  = 1|X1  = 0).

9. For two states x and y, we say that x → y if there exists some positive integer n (potentially depending on x, y) such that p(n)(x, y)  >  0.  For the gambler’s ruin chain on {0, ..., N} covered in lecture, identify the pairs of states (x, y) such that x → y .

10. For two states x and y, we say that x ↔ y if both x → y and y → x. For the gambler’s ruin chain on {0, ..., N}, find the pairs of states (x, y) for which x ↔ y .