The question is: After n days, what is the distribution of wind directions among these days? This continuing process is called a Markov chain, discovered by Andrey Markov, a Russian mathematician during Shackleton’s time.
Here is the formal definition:
n X XXX ...... 3 2 1
,
,
A sequence of random variables:
has Markov property
the future and past states are conditionally independent, if given the present state.
Pr(
X
x
X
x
,....,
X
x
Pr( ) X
x
X
x
)
n
1
1
1
n
n
n
1
n
n
Interestingly, this possibility distribution can be calculated by a matrix algorithm process called the transition matrix.
.
.
.
50 250 250 050 80 150 025 0 075 0 90 . .
.
.
Pr
.
.
Each row represents present states and each column represents the next states. For example, on row 1 column 1, it represents the probability of going from S wind to S wind.
th
n )1 ( state, starting
The transition matrix above shows the distribution for
th n state:
Pr x x
from
n
n
1
th n state is North
th
n )3 ( state if the
What about the distribution of the
wind?
X x
( Pr 1 n x
Pr Pr)
n
n
3
2
2
2
X
x
Pr
Pr )Pr (
n
n
1
3 Pr
X
n
13
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