Row-stochastic transition matrices; Population row vector R and update RT; Repeated predictions and steady state; Effects of transition rates and initial populations; Model assumptions and limitations
Example 1
Use row-vector convention v(next)=vP with P=[[0.9,0.1],[0.3,0.7]]. A population row is [360,0]. Find the second entry after one transition.
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Answer: 36
- Multiply the row by the second column: 360×0.1=36.
Example 2
For P=[[0.8,0.2],[0.3,0.7]], row vector [230,0] becomes [184,46]. Find its first entry after the next transition.
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Answer: 161
- First entry =184×0.8+46×0.3=161.
Example 3
For row-stochastic P=[[0.8,0.2],[0.3,0.7]], find the stationary proportion in state A.
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Answer: 0.6
- Let π=(p,1−p). Stationarity gives p=0.8p+0.3(1−p), hence 0.5p=0.3 and p=0.6.