Which of the following are probability vectors?

Which of the following are stochastic matrices?

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.)

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.)

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?

Find the steady state probability vector for the transition matrix $\left[\begin{array}{cc}\hfill 0.5\hfill & \hfill 1\hfill \\ \hfill 0.5\hfill & \hfill 0\hfill \end{array}\right]$.

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.)

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.)

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 $x_3$.

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.