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-p^2 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 p should he or she assert so as to maximize the expected score?
1 Answers
Best Answer
Define random variable X that marks the score of a certain meteorologist at some day. If the says that it will rain (he says it with probability p*) and it really rains (with probability p), he gets score 1-(1-p)^2. Also, if he says that it will not rain and it really does not rain (with probability 1-p*) and it really does not rain (with probability 1-p), he get score 1-p^2. Hence
E(X)=p**(1-(1-p)^2)+(1-p*)*(1-p^2)
=2pp*-p*-p^2+1
We now have to analyse the function p mapsto E(X) and find its maximum on (0,1). We have that
(dE(X))/(dp)=2p*-2p=0
Left-> p=p*
So we have that the meteorologist maximezes it chances when p=p*, which is according to our intuition.