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a. Find the steady-state vector for the Markov chain in Exercise 1 .b. At some time late in the day, what fraction of the listeners will be listening to the news?

(a) $x = \left[ \begin{array} { l } { \frac { 2 } { 3 } } \\ { \frac { 2 } { 3 } } \end{array} \right]$(b) $\frac { 2 } { 3 }$ are listening news at some time of the day.

Calculus 3

Chapter 4

Vector Spaces

Section 9

Applications to Markov Chains

Vectors

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in this video, we have a stochastic matrix p, which is provided here, and our first goal is to find the steady state vector of this matrix P. So to that end, what we need to do for the first step is to solve the equation that p times a vector X results in a vector X. This can be rearranged into the matrix equation p minus the identity matrix Times X equals E zero vector. Let's take this system here and go to an augmented matrix will be of size to buy three and from the Matrix p p minus. I tells us that we're subtracting one from the main diagonal. So here and here, that means will have a negative 0.3. Then the rest of the row is copy 0.6 and zero. Since we're augmenting with the zero vector, then copy points three into the second row and now take 0.4 minus one for a negative 10.6 and copy a zero. So this is the occupant in matrix for that matrix equation. And notice immediately that the second row is equal to the first row times negative one. That means in real reduction we can immediately wipe out Row two, then in Row one. If we divide by negative, 10.3 will obtain one negative to zero, and this matrix is already in reduce row echelon form. This implies that X one is going to be equal to two times x two x two is a free variable saw right X to his ex too. And now the solution for X from P X equals X is going to be if we pick a one for X two, we'll have a two x two in the first entry, which will be a two. This is not the steady state vector, but it's very close. We just need to multiply it by the right constant. If we take the some of the entries of X two plus one, resulting in three, then we can conclude that Q will be equal to 1/3 of the Vector X that we've just found. So this is going to be equal to 2/3 and 1/3. Let's pause for a moment and described what the steady state vector Q is indicating for us if we focus on the first entry 2/3 which is consistent with transitioning to news. We find the following that after many transitions, 2/3 of the viewers or approximately 2/3 we'll be listening to the news. So this is how to find the steady set victor Q as well as a bit of its interpretation.

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