3.18 Use first-step analysis to find the expected return time to state b for the Markov chain with transition matrix a b c a (1/2 1/2 P = b 1/4 3/4 c (1/2 1/2
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
please answer 3.18 use the example 3.17 method
![**3.18** Use first-step analysis to find the expected return time to state \( b \) for the Markov chain with transition matrix
\[
P = \begin{pmatrix}
1/2 & 1/2 & 0 \\
1/4 & 0 & 3/4 \\
1/2 & 1/2 & 0
\end{pmatrix}
\]
The matrix \( P \) is a transition matrix for a Markov chain with states \( a \), \( b \), and \( c \). The elements of the matrix represent the probabilities of moving from one state to another in a single step:
- From state \( a \):
- Probability of moving to \( a \): \( 1/2 \)
- Probability of moving to \( b \): \( 1/2 \)
- Probability of moving to \( c \): \( 0 \)
- From state \( b \):
- Probability of moving to \( a \): \( 1/4 \)
- Probability of moving to \( b \): \( 0 \)
- Probability of moving to \( c \): \( 3/4 \)
- From state \( c \):
- Probability of moving to \( a \): \( 1/2 \)
- Probability of moving to \( b \): \( 1/2 \)
- Probability of moving to \( c \): \( 0 \)
The goal is to calculate the expected return time to state \( b \) using first-step analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba5a3306-2973-4825-a0c7-f3a4b662c857%2Feb3424a2-fe47-47cb-95d8-f4838b5e08f2%2Fctkbdjj_processed.png&w=3840&q=75)
![**Example 3.17: Consider a Markov chain with transition matrix**
\[ P = \begin{bmatrix}
0 & 1 & 0 \\
1/2 & 0 & 1/2 \\
1/3 & 1/3 & 1/3
\end{bmatrix} \]
From state \( a \), find the expected return time \( E(T_a|X_0 = a) \) using first-step analysis.
**Solution:**
Let \( e_x = E(T_a|X_0 = x) \), for \( x = a, b, c \). Thus, \( e_a \) is the desired expected return time, and \( e_b \) and \( e_c \) are the expected first passage times to \( a \) for the chain started in \( b \) and \( c \), respectively.
For the chain started in \( a \), the next state is \( b \), with probability 1. From \( b \), the further evolution of the chain behaves as if the original chain started at \( b \). Thus,
\[ e_a = 1 + e_b. \]
From \( b \), the chain either hits \( a \), with probability 1/2, or moves to \( c \), where the chain behaves as if the original chain started at \( c \). It follows that
\[ e_b = \frac{1}{2} + \frac{1}{2}(1 + e_c). \]
Similarly, from \( c \), we have
\[ e_c = \frac{1}{3} + \frac{1}{3}(1 + e_b) + \frac{1}{3}(1 + e_c). \]
Solving the three equations gives
\[ e_c = \frac{8}{3}, \quad e_b = \frac{7}{3}, \quad \text{and} \quad e_a = \frac{10}{3}. \]
The desired expected return time is \( 10/3 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba5a3306-2973-4825-a0c7-f3a4b662c857%2Feb3424a2-fe47-47cb-95d8-f4838b5e08f2%2Fg9t8z8_processed.png&w=3840&q=75)
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