به انگلیسی

Tomorrow depends only on today

Mathematics · Probability · ages 22-24

Name ______________________ Date ____________

Sunny goes to Sunny with probability 0.8 and Cloudy with 0.2. Cloudy goes to Sunny with 0.4 and Cloudy with 0.6. Use order (Sunny, Cloudy) for both rows and columns. Which transition matrix fits?
- Rows (0.8, 0.4) and (0.2, 0.6)
- Rows (0.8, 0.2) and (0.4, 0.6)
- Rows (0.2, 0.8) and (0.6, 0.4)
Why does each row of a finite-state transition matrix sum to 1?
- It covers all mutually exclusive next states.
- It requires all next states to be equally likely.
- It requires each column to have a sum of zero.
In a Markov chain, learning a state from three steps ago can change the next-state probabilities even after the current state is known.
Circle one: True False
Use state order (A,B) for rows and columns. A transition matrix has rows (0.6,0.4) and (0.35,0.65). What does the entry 0.35 describe?
- From A, next B has chance 0.35.
- From A, next A has chance 0.35.
- From B, next A has chance 0.35.
- From B, next B has chance 0.35.
A time-homogeneous chain has order (A,B) and transition rows (0.5,0.5), (0.25,0.75). Starting in B, what is the probability of following exactly the three-step path B, A, A, B?
Answer: ______________
Use order (A,B). The first-step matrix P has rows (0.6,0.4), (0.2,0.8). The second-step matrix Q has rows (0.9,0.1), (0.5,0.5). These are the conditional transitions for each step. Starting in A, what is the chance of A after both steps?
Answer: ______________
A weather rule gives next-day Sunny probability 0.75 if the last two days match and 0.25 if they differ. It uses no older days. Storing the ordered pair of last two days makes the pair process Markov.
Circle one: True False
A process on states A and B chooses its next state by a fresh fair coin at odd times and keeps its current state at even times. Because its transition probabilities change with time, it cannot be Markov.
Circle one: True False
A rule uses only the last two days. Tomorrow's Sunny probability is 0.9 if yesterday was Sunny and 0.3 if yesterday was Cloudy. Store pairs as (yesterday,today), with S = Sunny and C = Cloudy. From (S,C), which next-pair law fits?
- To (S,S): 0.9; to (S,C): 0.1
- To (C,S): 0.9; to (C,C): 0.1
- To (C,S): 0.3; to (C,C): 0.7
- To (C,S): 0.1; to (C,C): 0.9
A random walk has states 0, 1, 2. From either end it moves to 1 with probability 1. From 1 it moves to 0 or 2 with probability 0.5 each. Starting at 0, what is the probability of being at 1 after exactly two steps?
- 0.5
- 1
- 0

Answer key

For grown-ups. Fold this page away before handing over the rest.

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Rows (0.8, 0.2) and (0.4, 0.6) · The Sunny row lists 0.8 to Sunny and 0.2 to Cloudy. The Cloudy row lists 0.4 to Sunny and 0.6 to Cloudy.
It covers all mutually exclusive next states. · Fixing the current state leaves an exhaustive list of mutually exclusive next states. Their probabilities sum to 1.
False · Earlier states add no next-state information once the current state is known.
From B, next A has chance 0.35. · The entry 0.35 is in row B and column A. It gives the probability of moving from B to A.
0.0625 · Only the specified path is counted: 0.25 × 0.5 × 0.5 = 0.0625.
0.74 · Use P first and Q second: 0.6 × 0.9 + 0.4 × 0.5 = 0.74.
True · The pair determines whether the days match. It gives the next-day law and the next-pair update without needing older days.
False · The rule uses the known time and current state. Earlier states add no information. It is Markov but not time homogeneous.
To (C,S): 0.9; to (C,C): 0.1 · Yesterday is S, so tomorrow is S with probability 0.9. Today's C becomes the older entry of the next pair.
0 · The first step reaches 1. The second step must reach 0 or 2, so no two-step path ends at 1.