Question about probabilities?

Hello, i`d like to know why the probabilities From the image are:

Pr[Xk+1=1 | Xk=1]=0.5

Pr[Xk+1=2 | Xk=1]=0.5

Pr[Xk+1=1 | Xk=2]=0.2

Pr[Xk+1=2 | Xk=2]=0.8

Thank you so much

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  • jibz
    Lv 6
    8 months ago
    Favourite answer

    Empirically, the domain of the given Markov sequence X(1), ... , X(8) has just two elements (a.k.a. "states"): 1 and 2; so its transition matrix A will be 2×2. To construct A we inspect terms X(1), ... , X(7), noting that

    2 of them are equal to 1 and

    5 of them are equal to 2,

    then ask: How many of them...

    ...is equal to 1 and is succeeded by a 1? Answer:1 term, namely X(4).

    ...is equal to 1 and is succeeded by a 2? Answer:1 term, namely X(5).

    ...is equal to 2 and is succeeded by a 2? Answer:4 terms, namely X(1), X(2), X(6), and X(7),

    ...is equal to 2 and is succeeded by a 1? Answer:1 term, namely X(3)

    (NB our scope excluded X(8) because it doesn't have a successor.) So, to rehash using set notation where #• is used to denote the number of elements in •, we've worked out that

    #{k∈{1, ... , 7} : X(k)=1} = 2.

    #{k∈{1, ... , 7} : X(k)=2} = 5.

    #{k∈{1, ... , 7} : X(k)=1 and X(k+1)=1} = 1.

    #{k∈{1, ... , 7} : X(k)=1 and X(k+1)=2} = 1.

    #{k∈{1, ... , 7} : X(k)=2 and X(k+1)=2} = 4.

    #{k∈{1, ... , 7} : X(k)=2 and X(k+1)=1} = 1.

    Finally, recall that given any state a and b, we have

    P[X(k+1)=a | X(k)=b] = #{k∈{1, ... , 7} : X(k)=b and X(k+1)=a} / #{k∈{1, ... , 7} : X(k)=b},

    and

    entry A(a,b) = Pr[X(k+1) = a | X(k) = b].

    • ...Show all comments
    • jibz
      Lv 6
      8 months agoReport

      oops, typo. i meant to write P[X(k+1)=1 | X(k)=2] = 1/5. you're dividing by the number of entries equal to 2.

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