Hello guys

I consider a setup of an electrical circuit, which can be in six different stages i.e 0 to 5 depicting how much energy is consumed. It can move up in energy level only by one state per time unit and the time is discrete, also only the current state influence the next. Which is why I reckon it should be a finite state discrete-time Markov chain. Furthermore, if it doesn't move up it can either stay or move down to another state, of course, this probability is depended on the current state. The downward probability follows a binomial distribution.

I then have to fill out the transition probability matrix $\displaystyle T$ given that:

$\displaystyle P(X_{n+1}=1|X_n=0) = 0.52$

$\displaystyle P(X_{n+1}=2|X_n=1) = 0.03$

$\displaystyle P(X_{n+1}=3|X_n=2) = 0.1$

$\displaystyle P(X_{n+1}=4|X_n=3) = 0.95$

$\displaystyle P(X_{n+1}=5|X_n=4) = 0.42$

$\displaystyle P(X_{n+1}=5|X_n=5) = 0.288$

In addition to the probability of moving to a lower energy consumption state or staying given no increase are:

$\displaystyle 5 \downarrow = 0.722$

$\displaystyle 4 \downarrow = 0.100$

$\displaystyle 3 \downarrow = 0.036$

$\displaystyle 2 \downarrow = 0.067$

$\displaystyle 1 \downarrow = 0.011$

$\displaystyle 0 \downarrow = 0$

I can write up

$\displaystyle T = \begin{bmatrix} ? & 0.52 & 0 & 0 & 0 & 0 \\ ? & ? & 0.03 & 0 & 0 & 0 \\ ? & ? & ? & 0.1 & 0 & 0 \\ ? & ? & ? & ? & 0.95 & 0 \\ ? & ? & ? & ? & ? & 0.42 \\ ? & ? & ? & ? & ? & 0.288 \end{bmatrix}$

I know that each row has to sum to one and that $\displaystyle T_{ij} > 0$, still I can't fill out the remaining fields.