<s>[INST] Chapter 2 Exercise 1 Pages 83 Linda J. S. Allen 2010 Exercises for Chapter 2 Exercise 1 Pages 83
textbook: An Introduction to Stochastic Processes with Applications to Biology 2nd Edition Linda J. S. Allen 2010
Link to the textbook

My attempt:
$P=(p_{ij})$ where $p_{ij}=\rm{Prob}\left\{X_{n+1}=i\lvert\ X_{n}=j\right\}$ is The one-step transition probability.
and $P$ is The transition matrix of the DTMC. also The transition matrix P is a stochastic matrix (A nonnegative matrix with the property that each column sum equals one is called a stochastic matrix.)
a stochastic matrix $\iff \begin{cases} \forall i,j\;  p_{ij}\geq 0 \\ \forall j;\; \sum_{i=1}^{\infty}p_{ij}=1 \end{cases}$
$P= \begin{pmatrix}
p_{11} & p_{12} & p_{13} \\
p_{21} & p_{22} & p_{23} \\
p_{31} & p_{32} & p_{33} 
\end{pmatrix}= \begin{pmatrix}
1 & 0 & 0 \\
\frac{1}{2} & 0 & \frac{1}{3} \\
0 & \frac{1}{3} & 1 
\end{pmatrix}$
but I don't feel I am right since the sum of each column doesn't equal 1
Thanks in adavance
 [/INST] $P= \begin{pmatrix}
p_{11} & p_{12} & p_{13} \\
p_{21} & p_{22} & p_{23} \\
p_{31} & p_{32} & p_{33} 
\end{pmatrix}= \begin{pmatrix}
a & 0 & 0 \\
\frac{1}{2} & 0 & \frac{1}{3} \\
0 & b & c 
\end{pmatrix}$
Since $\forall j\in [\![1,3]\!];\; \sum_{i=1}^{3}p_{ij}=1$, then it suffices to solve the following system of equations:
$\begin{cases} a+\frac{1}{2}+0=1 \\ 0+0+b=1 \\ 0+\frac{1}{3}+c=1 \end{cases}\implies \begin{cases} a=\frac{1}{2} \\ b=1 \\ c=\frac{2}{3} \end{cases}$
Thus the corresponding stochastic matrix
$$\boxed{P=\begin{pmatrix} \frac12 & 0 & 0 \\ \frac12 & 0 & \frac13 \\ 0 & 1 & \frac23  \end{pmatrix}}$$
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