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Math Help - Check if everything is OK in the way I have solved this problem?

  1. #1
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    Check if everything is OK in the way I have solved this problem?

    Can you please just look this over like peer review? I think I am fairly confident about my general procedure, but I am scared that I have committed minor errors. Thank you for your help.


    Suppose a gene in a chromosome is of type A or type B. Assume that the probability that a gene of type A will mutate to type B in one generation is 10^-4 and that a gene of type B will mutate to type A is 10^-6.

    (1) What is the transition matrix?

    10^-4 = 1/10^4 = 0.0001

    10^-6 = 1/10^6 = 0.000001

    A B
    A 0.9999 0.0001
    B 0.000001 0.999999

    (2) In the long run, what is the probability that the gene will be of type A? Of type B? (Find the stationary matrix.)

    [s1 s2] [0.9999 0.0001] = [s1 s2]
    [0.000001 0.999999]

    [0.9999 s1 + 0.000001 s2 0.0001 s1 + 0.999999 s2] = [s1 s2]

    Linear system


    0.9999 s1 + 0.000001 s2 = s1
    0.0001 s1 + 0.999999 s2 = s2
    s1 + s2 = 1

    -0.0001 s1 + 0.000001 s2 = 0
    0.0001 s1 - 0.000001 s2 = 0
    s1 + s2 = 1

    Algebraically solving for s1 and s2 using the equations from the linear system

    0.0001 s1 - 0.000001 s2 = 0
    0.0001 s1 = 0 + 0.000001 s2
    0.0001 s1 = 0.000001 s2
    s1 = (0.000001 s2 / 0.0001)

    s1 + s2 = 1
    (0.000001 s2 / 0.0001) + s2 = 1
    0.000001 s2 + 0.0001 s2 = 0.0001
    0.000101 s2 = 0.0001
    s2 = 0.0001/0.000101
    s2 = 0.990099009

    s1 + s2 = 1
    s1 = 1 - 0.990099009 = 0.00990099

    S = [0.00990099 0.9900099009]

    In the long run, the probability that a gene will be type A is 0.0099009 and the probability that it will be of type B is 0.9900099009.
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  2. #2
    MHF Contributor
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    Re: Check if everything is OK in the way I have solved this problem?

    Hey Yoodle15.

    Using Octave we get the answers:

    >> A
    A =

    1.00000 1.00000 1.00000
    -0.00010 0.00000 0.00000
    0.00010 -0.00000 0.00000

    >> rref(A)
    ans =

    1.00000 0.00000 0.00990
    0.00000 1.00000 0.99010
    0.00000 0.00000 0.00000

    which give s1 = 0.00990 and s2 = 0.99010 which is extremely close to your answer (I'm guessing its got to do with round-off errors).
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