Markov chain

#1
hi

I'm struggling with this question:


(Genetics)

Two animals are mated. From their offspring, two are selected and they
are mated together, and the process repeated. Each animal posseses a certain
gene, which can be lablled 'a' or 'b'. Each animal has 2 such genes and
can therefore be classified as 'aa', 'bb' or 'ab'. In the process we can
assume random mating such that, for example:



This leads to the pair of selected offspring being also 'ab bb'
(same as the parents) with probability :

1/2 * 1/2 + 1/2 * 1/2 = 1/2


(Because we can have the first offspring being ab, and the second bb, OR
the first being bb and the second ab). From this, construct the transition
matrix for this Markov process, and classify all comminications and any
absorbing states.


[Hint: Consider the genetic classification of each mating pair as a state
there are 6 in all.]


I have attached the full question (along with my workings) since the formatting wasn't too good.

How do i consruct the the transition matrix??


any help appreciated as i am really struggling with this.


thanks in advance


After some thought if i list all possible states:

1. aa,aa
2. aa,ab
3. aa,bb
4. ab,ab
5. ab,bb
6. bb,bb


(so 6 states) then this fits in with the hint.


now we need a transition matrix A, where A_i,j is the probability of ending in state i given we start in state j.


from 4: ab,ab, possibilities for offspring are: aa,bb or ab and we have to repeat the process for all offspring. This gives us


1. aa,bb,ab
2. bb,aa,ab
3. ab,aa,bb
4. ab,bb,aa
5. aa,ab,bb
6. bb,ab,aa


the above possibilities.


am i right to say that the selected offspring being also ab,ab, is (1/2*1/2*1/2)+(1/2*1/2*1/2)+...+(1/2*1/2*1/2) six times, or 6(1/8)??


And will my matrix need to be a 6*6 with aa,aa,ab,ab,bb,aa for rows and aa,ab,ab,bb,bb,bb for cols (in any order)??

thanks




anyone??
 
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