# Best method to find the stationary distribution

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asked May 18, 2018
edited May 21, 2018

What is the best method to find the stationary distribution? I attempted solving all the simultaneous equations but ended up with the wrong answer. See Models Tut 5 Question 4 b iii)

## 1 Answer

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answered May 21, 2018 by (580 points)

Typically the stationary distributions will be relatively easy to find given that you find the correct simultaneous equations - however in more complex cases Gauss reduction can be used to find the correct solution.

The case of Tut 5 Q4 b iii) is solved as follows, where:

• 3a = Level 3 this year following level 2 last year;
• 3b = Level 3 this year following level 4 last year.

and the corresponding transition matrix isĀ

 1 2 3a 4 3b 1 0.2 0.8 0 0 0 2 0.2 0 0.8 0 0 3a 0 0.2 0 0.8 0 4 0 0 0 0.8 0.2 3b 0.2 0 0 0.8 0

hence, the simultaneous equations are as follows:

$$\pi_1 = 0.2\pi_1 + 0.2\pi_2 +0.2\pi_{3b} \\ \pi_2 = 0.8\pi_1 +0.2\pi_{3a} \\ \pi_{3a} = 0.2\pi_{3b} + 0.2\pi_{3a} \\ \pi_{3b} = 0.2\pi_4$$ and the final condition of $$\sum_{i=1}^5 \pi_i =1$$

Solving the above through either substitution, ensuring that we use our limiting condition $$\sum_{i=1}^5 \pi_i =1$$, or using Gauss reduction, we obtain $$(\pi_1, \pi_2,\pi_{3a},\pi_4,\pi_{3b}) = \frac{1}{441}(21,20,16,320,64)$$ and hence the long-run proportion of time spent in level 3 is $$\frac{16+61}{441} = 0.18141$$.