Quiz 1 for STAT3921/4021
April 6, 2020
Write down your name in the first page and your SID on every page.
One double-sided A4 sheet of hand written notes is permitted.
Write your answers clearly in ink, not in pencil.
Except explicitly mentioned, no justification is required.
NAME: ....................................................................................
SID: ....................................................................................
1
SID: ....................................................................................
1. 1 mark for each question.
Let
P =

1 2 3 4 5
1 1 0 0 0 0
2 α 0 β 0 0
3 0 1/3 0 2/3 0
4 0 0 0 0 1
5 0 0 0 1/2 1/2
,
where 0 < α < 1 and 0 < β < 1.
(a). Specify all values of α so that P is a transition matrix of some Markov Chain.
(b). Specify communicating classes of the Markov chain in (a).
(c). Determine recurrent and transient classes of the Markov chain in (a).
(d). Provide the probability P (X5 = 2 | X0 = 4).
(e). Find all periodic states.
(f). Find the mean recurrence time of state 2.
2
SID: ....................................................................................
2. 2 marks for each question. Short justification is required.
Let {Xn, n ≥ 0} be a branching process with offspring ξ10 having the distribution:
P (ξ10 = 0) = α and P (ξ10 = 2) = 1− α.
(a) Find all α so that the offspring variance σ2 is greater than the offspring mean
µ.
(b) Suppose that X0 = 1 and σ
2 > µ. Find the extinction probability q.
(c) Suppose that X0 = 1. Find all α so that the extinction probability q = 1/2.
3
SID: ....................................................................................
3. 3 marks. Justification is required.
Let {Xn, n ≥ 0} be a branching process with offspring ξ10 having the distribution:
P (ξ10 = 0) = α and P (ξ10 = 2) = 1− α.
Suppose α = 1/4 and X0 ∼ Possion(λ), where λ > 0. Prove the extinction
probability q = exp(−2λ/3).
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