Intelligent Systems II · TEE · papers

Q6(b) Markov property and weather path [10]

Final examination · 17 Apr 2024

Actual historical paper ·100 marks attempted /140 offered · Q1 compulsory + any four of Q2–Q7. Source defects and solution assumptions are explicitly identified.

Question. Q6(b) [10 marks · 2024-04-17]

Describe the concept of a Markov process and its use in probabilistic systems. For the given state diagram S, C, R: (a) today is sunny; find probability tomorrow is sunny and the next day rainy. (b) today is cloudy and yesterday rainy; find probability tomorrow is sunny.

Swipe across the diagram to read its labels ↔
Redrawn transition diagram. Each arrow goes from the current state to the next state; labels are probabilities.
Diagram: Redrawn transition diagram. Each arrow goes from the current state to the next state; labels are probabilities.. Labels: legacy24-weather | 0.8 | 0.05 | 0.15 | 0.75 | 0.2 | 0.05 | 0.38 | 0.02 | 0.6 | Sunny | Cloudy | Rainy
Redrawn transition diagram. Each arrow goes from the current state to the next state; labels are probabilities.

Answer

Given

State order Sunny, Cloudy, Rainy. The diagram gives P=[[.8,.05,.15],[.75,.2,.05],[.38,.02,.6]].

To find

Definition and both requested conditional probabilities.

Formula and why it applies

A Markov process has the memoryless property: after the current state is known, earlier states do not change the next-state probabilities. It models time evolution through a state distribution and transition law.

P(X_{t+1}\mid X_t,X_{t-1},\ldots)=P(X_{t+1}\mid X_t)LaTeX: P(X_{t+1}\mid X_t,X_{t-1},\ldots)=P(X_{t+1}\mid X_t)

Solution

Step 1. Read origin row and destination column
P=\begin{bmatrix}0.8&0.05&0.15\\0.75&0.2&0.05\\0.38&0.02&0.6\end{bmatrix}LaTeX: P=\begin{bmatrix}0.8&0.05&0.15\\0.75&0.2&0.05\\0.38&0.02&0.6\end{bmatrix}

For example P(Sunny→Rainy)=0.15 while P(Rainy→Sunny)=0.38; arrow directions cannot be interchanged. Each row sums to 1.

Step 2. Multiply along the specified S→S→R path
P(X_{t+1}=S,X_{t+2}=R\mid X_t=S)=P_{SS}P_{SR}=0.8\times0.15=0.12LaTeX: P(X_{t+1}=S,X_{t+2}=R\mid X_t=S)=P_{SS}P_{SR}=0.8\times0.15=0.12

Do not sum all two-day routes ending in rain: tomorrow is specifically required to be sunny.

Step 3. Use the current cloudy state in part (b)
P(X_{t+1}=S\mid X_t=C,X_{t-1}=R)=P_{CS}=0.75LaTeX: P(X_{t+1}=S\mid X_t=C,X_{t-1}=R)=P_{CS}=0.75

Yesterday’s rain adds no further information under the stated Markov model. Such models are used for weather regimes, brand switching, queue states and reliability; the chosen state must capture the relevant predictive information.

Final answer. (a) 0.12; (b) 0.75.