Thread: Does all entries in a matrix of transition probabilities sum to 1 ?

Does all entries in a matrix of transition probabilities sum to 1 ?

What is the definition of "matrix of transition probabilities"?

Not sure because I am trying to learn statistics (topic: Markov Process) and it has a statement which asks the reader if this it true or false "All the entries in a matrix of transition probabilities sum to 1"

You have a text book that asks questions about "Markov Processes" without telling you what a "Markov Process" is?

Oh I am sorry, I thought you are asking me why I can;t answer this when I know the definition of "matrix of transition probabilities".

Based on my understanding from the reading the chapter from book, this is what I got:

Markov Process: A model used to analyze the evolution of a system over repeated trials when the state of the system at a given time cannot be determined with certainty.
State (of the system): The condition of the system at any particular trial or time period.
Transition Probability: The probability the system will be in state j during time period n+1 given that the system is in state i during time period n.

Markov Chain with Stationary Transition Probabilities:
A Markov Process in which:
Finite number of states
Transition probabilities remain constant
Probability of being in a particular state at any time period depends only on the state in the immediately preceding time period (memoryless property).

Transition Probabilities-
pij = probability of making a transition from state i in one period to state j in the next period.
P = matrix of transition probabilities.

State Probability πi(n) = the probability the system is in state i during period n.
Π(n) = [π1(n) π2(n) … πm(n)] = vector of state probabilities in period n.

There is no exact definition for the term "matrix of transition probabilities" - but that's what the question is that I am trying to answer if this statement is true or not "Does all entries in a matrix of transition probabilities sum to 1 ?"

Originally Posted by mathlearn

There is no exact definition for the term "matrix of transition probabilities" - but that's what the question is that I am trying to answer if this statement is true or not "Does all entries in a matrix of transition probabilities sum to 1 ?"

It should be written somewhere in your book that each row of a transition probability matrix should sum up to 1.
If the sum of each row and the sum of each column is equal to one, then your matrix is called a double stochastic matrix

Yea that I know that sum of all elements in a row is 1... but sum of all entries in transition probability matrix can't be 1 right ?

Originally Posted by mathlearn

Yea that I know that sum of all elements in a row is 1... but sum of all entries in transition probability matrix can't be 1 right ?

each row sums upto 1. So for a process with multiple states, the sum of all the entries of the matrix cannot be 1

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Originally Posted by mathlearn

thank you for the explanation

Since we are talking about matrix I wanna ask - When adding a square matrix, A, to its identity matrix, I, the sum is equal to matrix A ? or not ?

add the two matrices and see

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