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Each night different meteorologists give us the
Chapter 4, Problem 31P(choose chapter or problem)
Problem 31P
Each night different meteorologists give us the probability that it will rain the next day. To judge how well these people predict, we will score each of them as follows: If a meteorologist says that it will rain with probability p, then he or she will receive a score of
1 - (1 - p)2 if it does rain
1 − p2 if it does not rain
We will then keep track of scores over a certain time span and conclude that the meteorologist with the highest average score is the best predictor of weather. Suppose now that a given meteorologist is aware of our scoring mechanism and wants to maximize his or her expected score. If this person truly believes that it will rain tomorrow with probability p*, what value of pshould he or she assert so as to maximize the expected score?
Questions & Answers
QUESTION:
Problem 31P
Each night different meteorologists give us the probability that it will rain the next day. To judge how well these people predict, we will score each of them as follows: If a meteorologist says that it will rain with probability p, then he or she will receive a score of
1 - (1 - p)2 if it does rain
1 − p2 if it does not rain
We will then keep track of scores over a certain time span and conclude that the meteorologist with the highest average score is the best predictor of weather. Suppose now that a given meteorologist is aware of our scoring mechanism and wants to maximize his or her expected score. If this person truly believes that it will rain tomorrow with probability p*, what value of pshould he or she assert so as to maximize the expected score?
ANSWER:
Solution
Step 1 of 1
We have to find the value of p that should maximise the expected score
Given that meteorologist truly believes that tomorrow it will rain with probability
Given that the score is if it rains
And the score is