MAS328 Solutions to the final exam. Question 1. (20 marks) Four ...

(ii) Calculate the long run probability distribution (π0, π1, π2) for Xt.
Solving for πQ = 0 we have
⎡
πQ = [π0, π1, π2] ⎣
⎡
−0.5 0.5 0
0.2 −0.45 0.25
0 0.4 −0.4
= ⎣
−0.5 × π0 + 0.2 × π1
0.5 × π0 − 0.45 × π1 + 0.4 × π2
0.25 × π1 − 0.4 × π2
= [0, 0, 0],
⎤
⎦
⎤
⎦
T
MAS328
i.e. π0 = 0.4 × π1 = 0.64 × π2 under the condition π0 + π1 + π2 = 1,
which gives π0 = 16/81, π1 = 40/81, π2 = 25/81.
(iii) Compute the average number of operating machines in the long run.
In the long run the average is
0 × π0 + 1 × π1 + 2 × π2 = 40/81 + 50/81 = 90/81.
(iv) If an operating machine produces 100 units of output per hour, what
is the long run output per hour of the system ?
We find
100 × 90/81 = 1000/9.
Question 3. (20 marks)
Families in a distant society continue to have children until the first girl, and
then cease childbearing. Let X denote the number of male offsprings of a
particular husband.
(i) Assuming that each child is equally likely to be a boy or a girl, give the
probability distribution of X.
We have P(X = k) = (1/2) k+1 , k ∈ N.
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