## MATH1014 Quizzes

Quiz 9: Markov Chains and Leslie Matrices
Question 1 Questions
Which of the following are probability vectors? (Zero or more options can be correct)
 a) $\left[\begin{array}{c}\hfill 0.5\hfill \\ \hfill 0.5\hfill \end{array}\right]$ b) $\left[\begin{array}{c}\hfill 0.5\hfill \\ \hfill 0.5\hfill \\ \hfill 0.5\hfill \end{array}\right]$ c) $\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \\ \hfill 0\hfill \end{array}\right]$ d) $\left[\begin{array}{c}\hfill \frac{1}{6}\hfill \\ \hfill \hfill \\ \hfill \hfill \\ \hfill \frac{1}{2}\hfill \\ \hfill \hfill \\ \hfill \hfill \\ \hfill \frac{1}{3}\hfill \end{array}\right]$ e) $\left[\begin{array}{c}\hfill 0.72\hfill \\ \hfill 0.28\hfill \end{array}\right]$ f) $\left[\begin{array}{c}\hfill 1.3\hfill \\ \hfill -0.7\hfill \\ \hfill 0.4\hfill \end{array}\right]$

There is at least one mistake.
For example, choice (a) should be True.
There is at least one mistake.
For example, choice (b) should be False.
There is at least one mistake.
For example, choice (c) should be True.
There is at least one mistake.
For example, choice (d) should be True.
There is at least one mistake.
For example, choice (e) should be True.
There is at least one mistake.
For example, choice (f) should be False.
Correct!
1. True
2. False
3. True
4. True
5. True
6. False
Which of the following are stochastic matrices? (Zero or more options can be correct)
 a) $\left[\begin{array}{cc}\hfill 0.4\hfill & \hfill 0.3\hfill \\ \hfill 0.6\hfill & \hfill 0.7\hfill \end{array}\right]$ b) $\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ c) $\left[\begin{array}{cc}\hfill 0.4\hfill & \hfill 0.6\hfill \\ \hfill 0.3\hfill & \hfill 0.7\hfill \end{array}\right]$ d) $\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right]$ e) $\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ f) $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill \frac{1}{2}\hfill & \hfill \frac{1}{3}\hfill \\ \hfill \hfill \\ \hfill \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{3}\hfill \\ \hfill \hfill \\ \hfill \hfill \\ \hfill 0\hfill & \hfill \frac{1}{2}\hfill & \hfill \frac{1}{3}\hfill \end{array}\right]$ g) $\left[\begin{array}{ccc}\hfill \frac{1}{3}\hfill & \hfill \frac{1}{3}\hfill & \hfill \frac{1}{2}\hfill \\ \hfill \hfill \\ \hfill \hfill \\ \hfill \frac{1}{6}\hfill & \hfill \frac{1}{3}\hfill & \hfill -\frac{1}{2}\hfill \\ \hfill \hfill \\ \hfill \hfill \\ \hfill \frac{1}{2}\hfill & \hfill \frac{1}{3}\hfill & \hfill 1\hfill \end{array}\right]$

There is at least one mistake.
For example, choice (a) should be True.
There is at least one mistake.
For example, choice (b) should be True.
There is at least one mistake.
For example, choice (c) should be False.
There is at least one mistake.
For example, choice (d) should be True.
There is at least one mistake.
For example, choice (e) should be False.
There is at least one mistake.
For example, choice (f) should be True.
There is at least one mistake.
For example, choice (g) should be False.
All entries must be positive.
Correct!
1. True
2. True
3. False Columns must add to 1.
4. True
5. False Columns must add to 1.
6. True
7. False All entries must be positive.
Let $P=\left[\begin{array}{ccc}\hfill 0.3\hfill & \hfill 0.4\hfill & \hfill 0.25\hfill \\ \hfill 0\hfill & \hfill 0.4\hfill & \hfill 0.75\hfill \\ \hfill 0.7\hfill & \hfill 0.2\hfill & \hfill 0\hfill \end{array}\right]$ be the transition matrix for a Markov chain with 3 states. What is the probability that something in state 2 initially will be in state 3 at the next observation? (Enter your answer into the answer box.)

Correct!

The probability of moving from state 2 to state 3 is the $\left(3,2\right)$ entry of the transition matrix.
Let $P=\left[\begin{array}{ccc}\hfill 0.3\hfill & \hfill 0.4\hfill & \hfill 0.25\hfill \\ \hfill 0\hfill & \hfill 0.4\hfill & \hfill 0.75\hfill \\ \hfill 0.7\hfill & \hfill 0.2\hfill & \hfill 0\hfill \end{array}\right]$ be the transition matrix for a Markov chain with 3 states. What proportion of the initial state 1 population will be in state 2 after 2 steps? (Enter your answer into the answer box.)

Correct!

The answer required is the $\left(2,1\right)$ entry of ${P}^{2}$.
Suppose that Amy either jogs or rides her bike every day for exercise. If she jogs today, then tomorrow she will flip a fair coin and jog if it lands heads and ride her bike if it lands tails. If she rides her bike one day, then she will always jog the next day. This situation can be modelled as a Markov chain with 2 states. Taking“jog” to be state 1, and “bike ride” to be state 2, what is the transition matrix? Exactly one option must be correct)
 a) $\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right]$ b) $\left[\begin{array}{cc}\hfill 0.5\hfill & \hfill 1\hfill \\ \hfill 0.5\hfill & \hfill 0\hfill \end{array}\right]$ c) $\left[\begin{array}{cc}\hfill 0.5\hfill & \hfill 0\hfill \\ \hfill 0.5\hfill & \hfill 1\hfill \end{array}\right]$ d) $\left[\begin{array}{cc}\hfill 0.5\hfill & \hfill 0.5\hfill \\ \hfill 0.5\hfill & \hfill 0.5\hfill \end{array}\right]$ e) $\left[\begin{array}{cc}\hfill 0.5\hfill & \hfill 0.5\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right]$

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Find the steady state probability vector for the transition matrix $\left[\begin{array}{cc}\hfill 0.5\hfill & \hfill 1\hfill \\ \hfill 0.5\hfill & \hfill 0\hfill \end{array}\right]$. Exactly one option must be correct)
 a) $\left[\begin{array}{c}\hfill 2∕3\hfill \\ \hfill 1∕3\hfill \end{array}\right]$ b) $\left[\begin{array}{c}\hfill 1∕3\hfill \\ \hfill 2∕3\hfill \end{array}\right]$ c) $\left[\begin{array}{c}\hfill 0.5\hfill \\ \hfill 0.5\hfill \end{array}\right]$ d) $\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \end{array}\right]$ e) $\left[\begin{array}{c}\hfill 0\hfill \\ \hfill 1\hfill \end{array}\right]$

Choice (a) is correct!
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Let $L=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 0.8\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.7\hfill & \hfill 0\hfill \end{array}\right]$ be the Leslie matrix for an animal population with 3 age groups. On average, how many female offspring do females in the second age group produce? (Enter your answer into the answer box.)

Correct!

The required answer is the $\left(1,2\right)$ entry of $L$.
Let $L=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 0.8\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.7\hfill & \hfill 0\hfill \end{array}\right]$ be the Leslie matrix for an animal population with 3 age groups. Suppose that currently there are 100 females in the first age group, 60 in the second age group, and 50 in the third age group. How many females will be in the first age group in the next time period? (Enter your answer into the answer box.)

Correct!

Let ${x}_{0}=\left[\begin{array}{c}\hfill 100\hfill \\ \hfill 60\hfill \\ \hfill 50\hfill \end{array}\right]$. Find ${x}_{1}=L{x}_{0}$.
Let $L=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 6\hfill \\ \hfill \hfill \\ \hfill \hfill \\ \hfill \frac{1}{2}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill \\ \hfill \hfill \\ \hfill 0\hfill & \hfill \frac{1}{3}\hfill & \hfill 0\hfill \end{array}\right]$ be the Leslie matrix for a population, and let the initial population vector be ${x}_{0}=\left[\begin{array}{c}\hfill 100\hfill \\ \hfill 100\hfill \\ \hfill 100\hfill \end{array}\right]$. Find ${x}_{3}$. Exactly one option must be correct)
 a) $600$ b) $\left[\begin{array}{c}\hfill 600\hfill \\ \hfill 50\hfill \\ \hfill 100∕3\hfill \end{array}\right]$ c) $\left[\begin{array}{c}\hfill 200\hfill \\ \hfill 300\hfill \\ \hfill 50∕3\hfill \end{array}\right]$ d) $\left[\begin{array}{c}\hfill 100\hfill \\ \hfill 100\hfill \\ \hfill 100\hfill \end{array}\right]$

Choice (a) is incorrect
${x}_{3}$ is a $3×1$ matrix.
Choice (b) is incorrect
Your answer is ${x}_{1}.$
Choice (c) is incorrect
Your answer is ${x}_{2}.$
Choice (d) is correct!
A certain colony of lizards has a life span of less than 3 years. Suppose that the females are divided into 3 age groups: under age 1, age 1, and age 2. Suppose also that 50% of newborn females survive to age 1, and 30% of one-year-old females survive to age 2. Assume that females under age 1 do not give birth, while those of age 1 produce, on average, 1.2 female offspring and those of age 2 produce, on average, 2 female offspring. Write a Leslie matrix for this lizard population. Exactly one option must be correct)
 a) $\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill 50\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 30\hfill & \hfill 0\hfill \end{array}\right]$ b) $\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill 0.5\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.3\hfill & \hfill 0\hfill \end{array}\right]$ c) $\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 1.2\hfill & \hfill 2\hfill \\ \hfill 0.5\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0.3\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$ d) $\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 1.2\hfill & \hfill 2\hfill \\ \hfill 0.5\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.3\hfill & \hfill 0\hfill \end{array}\right]$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!