Stochastic

8 ab ab as a rst check on this formula we note that

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Unformatted text preview: NS To check this we note that 1/3 2/3 ✓ .6 .2 .4 .8 ◆ = ✓ .6 .4 + 3 3 .4 1.6 + 3 3 ◆ General two state transition probability. 1 2 1 1 a 1 b 2 a b We have written the chain in this way so the stationary distribution has a simple formula b a ⇡1 = ⇡2 = (1.8) a+b a+b As a first check on this formula we note that in the weather chain a = 0.4 and b = 0.2 which gives (1/3, 2/3) as we found before. We can prove this works in general by drawing a picture: b1 • a+b a2 !• b a a+b In words, the amount of sand that flows from 1 to 2 is the same as the amount that flows from 2 to 1 so the amount of sand at each site stays constant. To check algebraically that ⇡ p = ⇡ : b a b ba + ab b (1 a) + b= = a+b a+b a+b a+b b a ba + a ab a a+ (1 b) = = a+b a+b a+b a+b (1.9) Formula (1.8) gives the stationary distribution for any two state chain, so we progress now to the three state case and consider the Example 1.18. Social Mobility (continuation of 1.4). 1 2 3 1 .7 .3 .2 2 .2 .5 .4 3 .1 .2 .4 The equation ...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

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