What is the Transition Matrix?

Probability Level 1

A Markov chain has first state A and second state B, and its transition probabilities for all time are given by the following graph:

What is its transition matrix ?

Note: The transition matrix is oriented such that the k th k^\text{th} row represents the set of probabilities of transitioning from state k k to another state.

( 0.3 0.8 0.7 0.2 ) \begin{pmatrix} 0.3 & 0.8 \\ 0.7 & 0.2 \end{pmatrix} ( 1 0.8 0.7 1 ) \begin{pmatrix} 1 & 0.8 \\ 0.7 & 1 \end{pmatrix} ( 0.5 1.5 1.5 0.5 ) \begin{pmatrix} 0.5 & 1.5 \\ 1.5 & 0.5 \end{pmatrix} ( 0.3 0.7 0.8 0.2 ) \begin{pmatrix} 0.3 & 0.7 \\ 0.8 & 0.2 \end{pmatrix}

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Henry Maltby by Henry Maltby , 26, USA

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3 solutions

Henry Maltby
Apr 28, 2016

Relevant wiki: Markov Chains

Recall that the ( i , j ) th (i, \,j)^\text{th} entry of the transition matrix is Prob ( X n + 1 = j ∣ X n = i ) \text{Prob}(X_{n+1} = j \mid X_n = i) . Therefore, the rows each represent the outgoing arrows from a given vertex, and the answer is ( 0.3 0.7 0.8 0.2 ) . \boxed{\begin{pmatrix} 0.3 & 0.7 \\ 0.8 & 0.2 \end{pmatrix}.}

13 Helpful 3 Interesting 0 Brilliant 1 Confused

Since the total of transition probability from a state i i to all other states must be 1 1 , then

∑ j Pr ⁡ [ X n + 1 = j ∣ X n = i ] = 1. \sum_j \Pr\left[X_{n+1}=j\left|\right.X_{n}=i\right]=1.

The only possible answer is option 2.

Tunk-Fey Ariawan - 5 years, 1 month ago

Hey, you have to be careful there. There are two ways of handling with Matrices, since you can transpone a matrix, so that the rows and columns are exchanged.

Sandra gajic - 2 years, 1 month ago
Ash Gray
Nov 21, 2019

Parsing through the terse math we were given, it boils down to the explanation that for any state i the probability of transitioning to state j is given by the entry in the transition matrix at position i,j. Given that i indicates the row and j the column, we assume that a is the 0th position and b the 1st, so the transition probability from A to A is .3, A to B is .7, B to A is .8 and B to B is .2; therefore the correct matrix is ( . 3 . 7 . 8 . 2 ) \begin{pmatrix} .3 & .7 \\ .8 & .2 \end{pmatrix} .

5 Helpful 0 Interesting 0 Brilliant 0 Confused
Shekhar Koirala
Sep 30, 2018

( 0.65 0.35 ) ( 0.4 0.6 )

2 Helpful 1 Interesting 0 Brilliant 3 Confused

It's a trick question. The question asks for the Transition Matrix, not the 2 Step Matrix as demonstrated in the text above the question.

Andrew Aaron - 1 year, 5 months ago

1 pending report

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