IndE 311

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IndE 311 Problem Session Jan. 26th, 2007
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Pr. 16.2-2
a)

• Define states
• Let 0 the stock increased today and yesterday
• 1 the stock increased today but
  decreased yesterday
• 2 the stock decreased today but increased
  yesterday
• 3 the stock decreased today and yesterday
• Define Xt
• Xt the status of stock on day t (today) and day
  t-1 (yesterday)
• Define P

0 1 2 3
0 1 2 3
3
Pr. 16.2-2 (contd)
b) Because the state space is properly defined
as (change today, change yesterday), and
tomorrows state will have its second component
equal to todays first component. This captures
the probabilities given.

• Check
• finite number of states
• Markovian property
• Stationary transition property
• Initial probabilities

4
Pr. 16.2-3
a)

• Define states

State today yesterday 2 days ago
0 increase increase increase
1 increase increase decrease
2 increase decrease increase
3 increase decrease decrease
4 decrease increase increase
5 decrease increase decrease
6 decrease decrease increase
7 decrease decrease decrease

• Define Xt
• Xt the status of stock on day t, day t-1, and
  day t-2

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Pr. 16.2-3 (contd)
b) These states have included all the
information needed to predict tomorrows stock
market whereas the states in prob. 16.2-2 dont
consider the day before yesterday.
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Pr. 16.3-2
a)

• Define states
• 0 0 has been recorded
• 1 1 has been recorded
• Define Xt
• Xt binary digit recorded at the end of tth
  transmission

• Define P

0 1
0 1
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Pr. 16.3-2 (contd)
b)
0 1
0 1
The probability that a digit would be recorded
accurately is 0.909.
0 1
0 1
c)
0 1
0 1
The new probability is 0.99.
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Pr. 16.5-5
a)

• Define states
• 1 1 pint on hand just after a delivery
• 2 2 pints on hand just after a delivery
• Define Xt
• Xt number of pints after the tth delivery,
  every third day

7 7 pints on hand just after a delivery
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Pr. 16.5-5 (contd)

• Define P

1 2 3 4 5 6
7
1 2 3 4 5 6 7
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Pr. 16.5-5 (contd)
b)
c)
The steady-state probability that a pint of blood
is to be discarded
d) P(emergency deliveries)
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Pr. 5
a)

• Define states
• State (i,j,k) where
• i number of grants awarded (0 or 1) in year t-2
• j number of grants awarded (0 or 1) in year t-1
• k number of grants awarded (0 or 1) in year t
• Example If (i,j,k) (1,0,0) this year, and a
  grant is secured next year, the
• MC transitions to (i,j,k)
  (0,0,1)

• Define Xt
• Xt status of the owner at year t, t-1,
  and t-2

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Pr. 5 (contd)

• Define P

0-0-0 1-0-0 0-1-0 0-0-1 1-1-0 1-0-1
0-1-1 1-1-1
0-0-0 1-0-0 0-1-0 0-0-1 1-1-0 1-0-1 0-1-1 1-1-1
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Pr. 5(contd)
b)
Expected number of contracts in 3 years
State Steady-state
0-0-0 0.0148
1-0-0 0.0668
0-1-0 0.0668
0-0-1 0.0668
1-1-0 0.1783
1-0-1 0.1783
0-1-1 0.1783
1-1-1 0.2496
Expected number of contracts per year
14
Pr. 6
a)

• Define states
• State 1 person has last purchased cola 1
• State 2 person has last purchased cola 2
• Define Xt
• Xt The type of cola purchased by a person on
  her tth future cola purchase

• Define P

Cola 1 Cola 2
Cola 1 Cola 2
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Pr. 6 (contd)
Cola 1 Cola 2
Cola 1 Cola 2
The probability that she will purchase cola 1 in
two purchases after she purchased cola 2 is
b)
Cola 1 Cola 2
Cola 1 Cola 2
The probability that she will purchase cola 1 in
three purchases after she purchased cola 1 is