Question

1. (15 pts) A Markov chain $X_0, X_1, X_2, \dots$ has the transition probability matrix \begin{equation} \begin{array}{c|ccc} & 0 & 1 & 2 \\ \hline 0 & 0.3 & 0.2 & 0.5 \\ 1 & 0.5 & 0.1 & 0.4 \\ 2 & 0 & 0 & 1 \end{array} \end{equation} P = and is known to start in state $X_0 = 0$. Eventually, the process will end up in state 2. What is the probability that the time $T = \min\{n \ge 0; X_n = 2\}$ is an odd number?

          1. (15 pts) A Markov chain $X_0, X_1, X_2, \dots$ has the transition probability matrix
\begin{equation*}
\begin{array}{c|ccc}
 & 0 & 1 & 2 \\ \hline
0 & 0.3 & 0.2 & 0.5 \\
1 & 0.5 & 0.1 & 0.4 \\
2 & 0 & 0 & 1
\end{array}
\end{equation*}
P = 

and is known to start in state $X_0 = 0$. Eventually, the process will end up in state 2. What is
the probability that the time $T = \min\{n \ge 0; X_n = 2\}$ is an odd number?

1. (15 pts) A Markov chain X0, X1, X2, … has the transition probability matrix

0 1 2

0 0.3 0.2 0.5

1 0.5 0.1 0.4

2 0 0 1

P =

and is known to start in state X0 = 0. Eventually, the process will end up in state 2. What is
the probability that the time T = min{n ≥ 0; Xn = 2} is an odd number?

Added by Sergio S.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Robin Corrigan

01/28/2024

Step 1

Then, the probability that $T$ is odd is given by: $$ \sum_{n=1}^{\infty} p_{2n+1} $$ Show more…

Show all steps

Thanks for your feedback!

1. (15 pts) A Markov chain $X_0$, $X_1$, $X_2$, ... has the transition probability matrix $$ P = \begin{pmatrix} 0 & 1 & 2 \\ 0 & 0.3 & 0.2 & 0.5 \\ 1 & 0.5 & 0.1 & 0.4 \\ 2 & 0 & 0 & 1 \end{pmatrix} $$ and is known to start in state $X_0 = 0$. Eventually, the process will end up in state 2. What is the probability that the time $T = min\{n \ge 0; X_n = 2\}$ is an odd number?