 # Each night different meteorologists give us the probability that it wi leviattan0pi 2021-11-14 Answered
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
$$\displaystyle{1}-{\left({1}-{p}\right)}^{{2}}$$ if it does rain
$$\displaystyle{1}-{p}^{{2}}$$ 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 $$\displaystyle{p}\cdot$$, what value of p should he or she assert so as to maximize the expected score?

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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 $$\displaystyle{p}\cdot$$) and it really rains (with probability p), he gets score $$\displaystyle{1}-{\left({1}-{p}\right)}^{{2}}$$. Also, if he says that it will not rain and it really does not rain (with probability $$\displaystyle{1}-{p}\cdot$$) and it really does not rain (with probability 1-p), he get score $$\displaystyle{1}-{p}^{{2}}$$. Hence
$$\displaystyle{E}{\left({X}\right)}={p}\cdot\cdot{\left({1}-{\left({1}-{p}\right)}^{{2}}\right)}+{\left({1}-{p}\cdot\right)}\cdot{\left({1}-{p}^{{2}}\right)}$$
$$\displaystyle={2}{p}{p}\cdot-{p}\cdot-{p}^{{2}}+{1}$$
We now have to analyse the function $$\displaystyle{p}\mapsto{E}{\left({X}\right)}$$ and find its maximum on (0,1). We have that
$$\displaystyle{\frac{{{d}{E}{\left({X}\right)}}}{{{d}{p}}}}={2}{p}\cdot-{2}{p}={0}$$
$$\displaystyle\Leftrightarrow{p}={p}\cdot$$
So we have that the meteorologist maximezes it chances when $$\displaystyle{p}={p}\cdot$$, which is according to our intuition.