Each night different meteorologists give us the probability that it wil rain the

Question

Each night different meteorologists give us the probability that it wil 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 -p2if 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 this and so wants to maximize his or her expected score. If this person truly believes that it will rain tomorrow with probability q, what value of p should he or she assert so as to maximize the expected score?

Explanation / Answer

Solution: Here simply we have to find value of p.

We have to use probability distribution rule.

We know for any valid distribution, sum of probabilities equal to 1.

For first distribution, we calculate value of p.

(1-(1-p)^2)+(1-p^2) = 1

1(1p)^2+1p^2=1

Step 1: Simplify both sides of the equation.

2p^2+2p+1=1

Step 2: Subtract 1 from both sides.

2p^2+2p+11=11

2p^2+2p=0

Step 3: Factor left side of equation.

2p(p+1)=0

Step 4: Set factors equal to 0.

2p=0 or p+1=0

p=0 or p=1

Here choice of p is either 0 or 1.

For second distribution,

(1-(1-p)^2)+(1-0.9*p^2) = 1

1(1p)^2+10.9p^2=1

Step 1: Simplify both sides of the equation.

1.9p^2+2p+1=1

Step 2: Subtract 1 from both sides.

1.9p^2+2p+11=11

1.9p^2+2p=0

Step 3: Factor left side of equation.

p(1.9p+2)=0

Step 4: Set factors equal to 0.

p=0 or 1.9p+2=0

p=0 or p=1.052632

Here choice of p is either 0 or 1.052632

Here value of p>1 which is not acceptable.

Hence we consider correct choice of p = 0

Done

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