To tackle this question, we will break it down into parts:

Part 1: Show that $(a, b)$ is a fixed point of the stochastic matrix $P$.

The given matrix $P$ is:

$P = \begin{bmatrix} 1 - b & b \\ a & 1 - a \end{bmatrix}$

A fixed point of a stochastic matrix corresponds to a probability vector $\pi = (x, y)$ such that:

$\pi P = \pi$

We want to show that $(a, b)$ is a fixed point of $P$, meaning we need to prove that:

$(a, b) P = (a, b)$

Step 1: Set up the equation

Multiply the row vector $(a, b)$ by the matrix $P$:

$(a, b) \begin{bmatrix} 1 - b & b \\ a & 1 - a \end{bmatrix}$

This gives us two equations for the components of the vector:

$a(1 - b) + b a = a$ $a b + b(1 - a) = b$

Step 2: Simplify each equation

• For the first equation:

$a(1 - b) + b a = a$ Expanding both sides:

$a - ab + ab = a$

Both terms $-ab$ and $ab$ cancel out, leaving:

$a = a$

Thus, the first equation is satisfied.

• For the second equation:

$ab + b(1 - a) = b$ Expanding both sides:

$ab + b - ab = b$

The terms $ab$ and $-ab$ cancel out, leaving:

$b = b$

Thus, the second equation is satisfied.

Conclusion for Part 1:

We have shown that $(a, b)$ is indeed a fixed point of the matrix $P$. Therefore, $(a, b)$ is the fixed probability vector for this stochastic matrix.

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Part 2: Write down the fixed probability vectors for the given matrices

You provided the matrices $P_1$, $P_2$, and $P_3$, but it seems like there were some issues with the notation. Could you please clarify the specific matrices $P_1$, $P_2$, and $P_3$?

Once I have the correct matrix forms, I can proceed to compute their respective fixed probability vectors.

Would you like to clarify the matrices or proceed with the next steps?

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Five related questions:

1. What is a stochastic matrix, and how is it related to probability vectors?
2. How do you determine if a matrix has a fixed point?
3. Can a stochastic matrix have more than one fixed probability vector?
4. What are some applications of fixed points in probability and stochastic processes?
5. How does the stability of fixed points affect the behavior of stochastic systems?

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Tip:

When solving for the fixed probability vector of a stochastic matrix, remember that the rows of the matrix represent probability distributions, so each row must sum to 1.