SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...

67
= 1− Pr{X 1 >n} Pr{X 2 >n}
= 1− (Pr{X 1 >n}) 2
since spins are independent and identically distributed. Notice that Pr{X 1 >n} =
1 − p (n)
0,10. Then letting T be the total number of spins
Pr{T ≤ 2n} = Pr{Z ≤ n}
= 1− (1 − p 0,10) (n) which implies that
Pr{T =2n} = Pr{T ≤ 2n}−Pr{T ≤ 2(n − 1)}
= (1− p 0,10) (n) − (1 − p (n−1)
0,10 ) 2
This result can be used to calculate the distribution or expectation. You will find that
E[T ] < E[2X 1 ].
27. (a)
M = {1, 2, 3, 4, 5}
≡ {insert, withdraw, deposit, information, done}
n represents the number of transactions
⎛
P =
⎜
⎝
0 0.5 0.4 0.1 0
0 0 0.05 0.05 0.9
0 0.05 0 0.05 0.9
0 0.05 0.05 0 0.9
0 0 0 0 1
(b) Both properties are rough approximations at best. Customers are likely to do
only 1 of each type of transaction, so all of their previous transactions influence
what they will do next. Also, the more transactions they have made, the more
likely they are to be done, violating stationarity.
(c) Need f (n)
15 for n =1, 2,...,20 with A = {1, 2, 3, 4} and B = {5}.
⎞
⎟
⎠