Final answer:
A Discrete Time Markov Chain is defined for a system with weather states Sunny, Cloudy, and Rainy, and a Transition Probability Matrix is provided to represent the transition probabilities between these states.
Step-by-step explanation:
A Discrete Time Markov Chain (DTMC) is a stochastic model that describes a sequence of possible events where the probability of each event depends only on the state attained in the previous event. Let's consider a simple example involving three weather states: Sunny (S), Cloudy (C), and Rainy (R). The states here represent the weather conditions. We can represent the transitions between these states with a Transition Probability Matrix:
\[ P = \begin{bmatrix} P(SS) & P(SC) & P(SR) \\ P(CS) & P(CC) & P(CR) \\ P(RS) & P(RC) & P(RR) \end{bmatrix} \]
Assuming the weather follows a pattern where it remains the same with a probability of 0.5, changes from Sunny to Cloudy with a probability of 0.3, from Cloudy to Rainy with a probability of 0.2, and it's rare for weather to go directly from Sunny to Rainy or Rainy to Sunny, we may end up with a matrix like:
\[ P = \begin{bmatrix} 0.5 & 0.3 & 0.2 \\ 0.3 & 0.5 & 0.2 \\ 0.1 & 0.4 & 0.5 \end{bmatrix} \]
This matrix shows the probability of transitioning from one state to another. For example, P(SC) is the probability of transitioning from a Sunny day to a Cloudy day, denoted as 0.3 in our matrix.