Math Problem Statement
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Part 1
At one liberal arts college, students are classified as humanities majors, science majors, or undecided. The chances are
4040%
that a humanities major will change to a science major from one year to the next, and
2525%
that a humanities major will change to undecided. A science major will change to humanities with probability
0.350.35,
and to undecided with probability
0.450.45.
An undecided will switch to humanities or science with probabilities of
0.300.30
and
0.400.40,
respectively. Complete parts (a) and (b) below.
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Part 1
(a)**** Find the long-range prediction for the fraction of students in each of these three majors.
First, find the transition matrix P. Let the first state be that a student is a humanities major, the second that a student is a science major, and the third state that a student is undecided.
Pequals=Start 3 By 3 Table 1st Row 1st Column 0.35 2nd Column 0.40 3rd Column 0.25 2nd Row 1st Column 0.35 2nd Column 0.20 3rd Column 0.45 3rd Row 1st Column 0.30 2nd Column 0.40 3rd Column 0.30 EndTable
0.35
0.40
0.25
0.35
0.20
0.45
0.30
0.40
0.30
(Type an integer or decimal for each matrix element.)
Part 2
Find the long-range prediction for the fraction of students in each of these three majors.
The long-range prediction is
one third13
humanities,
one third13
science, and
one third13
undecided.
(Type integers or simplified fractions.)
Part 3
(b)**** For a regular 2-by-2 matrix having column sums of 1, the equilibrium vector is
Start 1 By 2 Table 1st Row 1st Column one half 2nd Column one half EndTable
12
12
.
Make a conjecture, and describe how this conjecture, if true, would allow you to predict the answer to part (a) with very little computation.
A regular 3-by-3 matrix having column sums of 1 has an equilibrium vector of
enter your response here.
(Type an integer or simplified fraction for each matrix element.)
Solution
Part 1(a): Transition Matrix
The problem describes the transitions between states (humanities, science, and undecided) as follows:
-
Humanities:
- Stays humanities:
- Changes to science:
- Changes to undecided:
-
Science:
- Changes to humanities:
- Stays science:
- Changes to undecided:
-
Undecided:
- Changes to humanities:
- Changes to science:
- Stays undecided:
The transition matrix is:
0.35 & 0.40 & 0.25 \\ 0.35 & 0.20 & 0.45 \\ 0.30 & 0.40 & 0.30 \end{bmatrix}$$ --- ### **Part 1(b): Long-range Prediction (Stationary Distribution)** To find the long-range prediction, solve for the stationary distribution $$\pi = [\pi_1, \pi_2, \pi_3]$$, such that $$\pi P = \pi$$ and $$\pi_1 + \pi_2 + \pi_3 = 1$$. From computations, the stationary distribution is: $$\pi = \left[\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right]$$ This means the **long-range prediction** is: - Humanities: $$\frac{1}{3}$$ (approximately 33.33%) - Science: $$\frac{1}{3}$$ (approximately 33.33%) - Undecided: $$\frac{1}{3}$$ (approximately 33.33%) --- ### **Part 1(c): Conjecture for a Regular 3x3 Matrix** For a **regular** $$3 \times 3$$ transition matrix where all column sums equal 1, the equilibrium vector (stationary distribution) has **equal components** if: 1. The system is symmetric in transitions, meaning no bias exists favoring one state over another. 2. All states are reachable, and the transitions stabilize over time. #### Conjecture: If $$P$$ is a **regular $$3 \times 3$$** matrix with column sums of 1, the equilibrium vector is: $$\pi = \left[\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right]$$ This conjecture applies due to the symmetry of the system, and it predicts the answer to **part (a)** with little computation. Specifically, each state will eventually hold an equal fraction of the population. --- ### Final Answers: - Transition matrix $$P$$: $$P = \begin{bmatrix} 0.35 & 0.40 & 0.25 \\ 0.35 & 0.20 & 0.45 \\ 0.30 & 0.40 & 0.30 \end{bmatrix}$$ - Long-range prediction (stationary distribution): $$\left[\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right]$$ - Conjecture for a regular $$3 \times 3$$ matrix: Stationary distribution is $$\left[\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right]$$. --- ### Questions to Explore: 1. Why is the stationary distribution equal for this particular transition matrix? 2. What happens if one state becomes unreachable? How does the equilibrium change? 3. How can we quickly identify whether a matrix is regular without detailed calculations? 4. What is the relationship between eigenvalues and stationary distributions in Markov processes? 5. How do the transition probabilities affect the speed of convergence to equilibrium? ### Tip: For a regular $$n \times n$$ Markov matrix where all states are symmetric and reachable, the stationary distribution is $$\frac{1}{n}$$ for each state. Always check for symmetry and regularity when analyzing such problems!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability
Markov Chains
Stationary Distribution
Transition Matrices
Formulas
Transition Matrix: Rows represent the probability of moving between states.
Stationary Distribution: πP = π and π₁ + π₂ + ... + πₙ = 1
Theorems
Markov Chain Convergence Theorem
Suitable Grade Level
Grades 10-12, College-level introductory math
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