Statistics - MathematicalStatistics

Question 1: The green represents success, red failure, what is the probability of reaching the success node starting at node 1?

1 1/4 0 1/2

Question 2: What is the probability of reaching the success node starting at node 2?

1 1/2 1/4 0

Question 3: What is the probability of reaching the success node starting at node 0?

1 0 1/2 1/4

Question 4: Consider the same situation but with n nodes pictured here, what is the probability of reaching the success node from node n?

0 1 1/2 1/4

Question 5: What is the probability of reaching the success node from node 0 in the new n walk?

0 1 1/2 1/4

Question 6: What is the probability of reaching the success node from node j? (Hint: use recursion)

0 1 1-(j/n) j/n

Question 7: How many steps will it take to finish the walk starting from node j?

j(n-j) n-j $j$ n(n-j)

Question 8: What is the expected number of steps to reach the success node from node j in 1 attempt?

j(n-j) 0 $\frac{j(n-j)}{2}$ undefined

Question 9: Is the expected number of steps to succeed or fail the sum of the expectation to fail, and succeed?

Only if the probabilities are 0 Only if the probabilities are 1 No, it is not the sum of the expectations Yes, it is the sum of the expectations

Question 10: Explain the previous response

The expectation is only the sum when they have high probability The expectation of steps is the sum of the 2 because they are the only 2 end nodes The expectation of steps has no information about which node it reaches The expectation is only the sum when they have low probability

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