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I came across this question: In the ultrastrators civilization on planet Anachronista, the population is divided into four strata which, in order of status , are labeled Alpha, Betta, Gamma and Delta. By the traditions of the civilization, no child can have a status more than one different from its parents. As examples, in each generation 20% of the children of Alphas grow up to be Bettas, the rest remaining Alphas, while of the Betts offspring 50% remain Bettas while 10% become Alphas and the rest Gammas.

I am asked to find the transitional matrix, per my understanding, the transitional matrix will be a four by four matrix since the population is divided into four strata. I am able to compute the first row [0.8, 0.2, 0, 0] and the second row [ 0.1, 0.5, 0.4, 0]. How will I find the remaining two rows to complete the transitional matrix?

Astatine
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You do not have sufficient information to uniquely determine the transition probability yet.

The following information are still needed:

  • Percentage of Gammas that will remain Gammas, percentage of Gammas that will become Betas.
  • Percentage of Deltas that will remain Deltas.
Siong Thye Goh
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  • I thought similar, but I was given the matrix in this format: the first row [ x, 0.2, y, 0] the second row [0.1, 0.5, a, b] the third row [C, d, 0.5, 0.4] and the fourth [ e, 0, g, 0.8]. I am able to find the x = 0.8 and y = 0. a =0.4. b = 0. Based on this format, can the remaining variables be found ? – Astatine Nov 05 '18 at 01:19
  • of course, each row sum to $1$. also make use of the information that no child can have a status more than one different from its parents. – Siong Thye Goh Nov 05 '18 at 01:24
  • Based on the statement no child can have a status more than one different from its parents. How will you find the remaining variables? – Astatine Nov 05 '18 at 10:41
  • is it possible for a gamma have a child who is alpha over one generation? what can you say about the value of $c$? – Siong Thye Goh Nov 05 '18 at 11:32
  • Based on the matrices given, it's possible to find the value of c and d. Since they are probabilities, I know each row sum up to 1. But don't know wether c = 0.1 or d = 0.1 – Astatine Nov 05 '18 at 12:00
  • $c$ represent the probability of a gamma becoming alpha over one generation, it shouldn't be possible according to the condition given right? so can you conclude what is the value of $c$? – Siong Thye Goh Nov 05 '18 at 12:07