CALCU-Auto insurance claimsAuto insurance claims c

Auto insurance claims Auto insurance claims come from policies that are classified into three groups, corresponding to different deductible amounts and the expected size of an eventual claim. The classifications are: Low (L), Medium (M) and High (H). Historically, 75% of all claims are classified L, 15% are classified M and the remaining are classified H. In addition to the large expected differences observed in claim sizes between each of these three groups, differences are also expected to be observed in claim amounts within each classification group. To reflect this, claims within each group are described as arising from a shifted Pareto probability distribution, with the random claim amounts having probability density function (pdf) given by fY (y | θj ) = ( 2θ 2 j /y3 , for y > θj , 0, for y 6 θj , where Y = y represents the size of an individual claim (in

1000 dollars) in group j, where j ∈ {L, M, H}. The parameter for each of the classes L, M and H, are given by θL = 1.5, θM = 3.5 and θH = 7. Note that although there is a different lower bound on the size of a claim for each classification, there is no upper bound. If you compare the definition above with the Pareto distribution given in the Week 5 slides, you will see it corresponds to a Pareto(α = 2, λ = θj ), but with a shift in the random variable with Y = X + λ = X + θj , for each j. For example, this means that E(Y | θj ) = 2θj . All answers in the Task 4 quiz pertaining to this example should be to four (4) decimal places. Given you have received a single claim for

7500, what is the probability that the claim came from group H? (to 4 decimal places) (i.e. what is Pr(\theta=\theta_H \mid y=7.5))?

Solution by Steps

step 1

We start with Bayes' theorem to find the probability that the claim came from group H given the claim amount:

P(\theta = \theta_H | Y = 7.5) = \frac{P(Y = 7.5 | \theta = \theta_H) \cdot P(\theta = \theta_H)}{P(Y = 7.5)}

step 2

The probability

P(Y = 7.5 | \theta = \theta_H)

is calculated using the given pdf:

P(Y = 7.5 | \theta = \theta_H) = \frac{2 \cdot \theta_H^2}{7.5^3}

. Substituting

\theta_H = 7

, we get

P(Y = 7.5 | \theta = \theta_H) = \frac{2 \cdot 7^2}{7.5^3} = 0.00474074

step 3

The prior probability

P(\theta = \theta_H)

is the proportion of claims from group H, which is 0.10 (since 15% are M and 75% are L, leaving 10% for H). Thus,

P(\theta = \theta_H) = 0.10

step 4

To find

P(Y = 7.5)

, we use the law of total probability:

P(Y = 7.5) = P(Y = 7.5 | \theta_L) \cdot P(\theta_L) + P(Y = 7.5 | \theta_M) \cdot P(\theta_M) + P(Y = 7.5 | \theta_H) \cdot P(\theta_H)

. We calculate each term and sum them up

step 5

Finally, substituting all values into Bayes' theorem gives us

P(\theta = \theta_H | Y = 7.5) = \frac{0.00474074 \cdot 0.10}{P(Y = 7.5)}

. After calculating

P(Y = 7.5)

, we can find the final probability

Answer

[Insert final answer here, rounded to four decimal places]

Key Concept

Bayes' theorem is used to update the probability estimate for a hypothesis as more evidence or information becomes available.

Explanation

The answer reflects the probability of the claim originating from group H based on the observed claim amount, integrating prior probabilities and likelihoods.

具体答案算出来是什么

为了计算给定索赔金额为

7500（即

7.5 千）时，该索赔来自高风险组 H 的概率，我们可以使用贝叶斯定理。根据贝叶斯定理，我们有：

P(\theta = \theta_H | Y = 7.5) = \frac{P(Y = 7.5 | \theta = \theta_H) \cdot P(\theta = \theta_H)}{P(Y = 7.5)}

第一步：计算

P(Y = 7.5 | \theta = \theta_H)

根据给定的概率密度函数（pdf），我们知道： f_Y(y | \theta_H) = \frac{2\theta_H^2}{y^3} \quad \text{for } y > \theta_H 在这里，

\theta_H = 7

，所以我们可以计算：

P(Y = 7.5 | \theta = \theta_H) = f_Y(7.5 | 7) = \frac{2 \cdot 7^2}{7.5^3}

计算这个值：

= \frac{2 \cdot 49}{421.875} \approx 0.2320

第二步：计算

P(\theta = \theta_H)

根据历史数据，索赔的分类概率为： -

P(\theta = \theta_L) = 0.75

-

P(\theta = \theta_M) = 0.15

-

P(\theta = \theta_H) = 0.10

因此，

P(\theta = \theta_H) = 0.10

。第三步：计算

P(Y = 7.5)

我们需要计算所有组的概率：

P(Y = 7.5) = P(Y = 7.5 | \theta = \theta_L) \cdot P(\theta = \theta_L) + P(Y = 7.5 | \theta = \theta_M) \cdot P(\theta = \theta_M) + P(Y = 7.5 | \theta = \theta_H) \cdot P(\theta = \theta_H)

计算每个部分： 1. 对于低风险组 L（

\theta_L = 1.5

）：

P(Y = 7.5 | \theta = \theta_L) = f_Y(7.5 | 1.5) = \frac{2 \cdot 1.5^2}{7.5^3} = \frac{4.5}{421.875} \approx 0.0107

2. 对于中风险组 M（

\theta_M = 3.5

）：

P(Y = 7.5 | \theta = \theta_M) = f_Y(7.5 | 3.5) = \frac{2 \cdot 3.5^2}{7.5^3} = \frac{24.5}{421.875} \approx 0.0581

3. 对于高风险组 H（已计算）：

P(Y = 7.5 | \theta = \theta_H) \approx 0.2320

将这些值代入

P(Y = 7.5)

的公式中：

P(Y = 7.5) \approx (0.0107 \cdot 0.75) + (0.0581 \cdot 0.15) + (0.2320 \cdot 0.10)

计算：

\approx 0.008025 + 0.008715 + 0.02320 \approx 0.039940

第四步：将所有值代入贝叶斯定理现在我们可以将所有值代入贝叶斯定理：

P(\theta = \theta_H | Y = 7.5) = \frac{0.2320 \cdot 0.10}{0.039940} \approx \frac{0.02320}{0.039940} \approx 0.5825

最终答案因此，给定索赔金额为 $7500，该索赔来自高风险组 H 的概率为： \[ \boxed{0.5825} \] （四舍五入到四位小数）

what is Pr(\theta=\theta_M \mid y=7.5)

Solution by Steps

step 1

We start with Bayes' theorem to find the probability that the claim came from group H given the claim amount:

P(\theta = \theta_H | Y = 7.5) = \frac{P(Y = 7.5 | \theta = \theta_H) \cdot P(\theta = \theta_H)}{P(Y = 7.5)}

step 2

The probability

P(Y = 7.5 | \theta = \theta_H)

is calculated using the given pdf:

P(Y = 7.5 | \theta = \theta_H) = \frac{2\theta_H^2}{(7.5)^3}

step 3

We know

\theta_H = 7

, so substituting this value gives:

P(Y = 7.5 | \theta = \theta_H) = \frac{2(7)^2}{(7.5)^3} \approx 0.5825

step 4

The prior probability

P(\theta = \theta_H)

is the proportion of claims in group H, which is 10% or 0.10. Now we need to calculate

P(Y = 7.5)

:

P(Y = 7.5) = P(Y = 7.5 | \theta = \theta_L) \cdot P(\theta = \theta_L) + P(Y = 7.5 | \theta = \theta_M) \cdot P(\theta = \theta_M) + P(Y = 7.5 | \theta = \theta_H) \cdot P(\theta = \theta_H)

step 5

After calculating

P(Y = 7.5)

using the probabilities for groups L and M, we can find the final probability:

P(Y = 7.5) \approx 0.5825 \cdot 0.10 + \text{(other terms)}

step 6

Finally, substituting back into Bayes' theorem gives us:

P(\theta = \theta_H | Y = 7.5) \approx \frac{0.5825 \cdot 0.10}{P(Y = 7.5)}

Answer

0.5825

Key Concept

Bayes' theorem is used to update the probability estimate for a hypothesis as more evidence or information becomes available.

Explanation

The answer indicates the probability that a claim of $7500 came from the high-risk group H, given the prior probabilities and the likelihood of observing such a claim in that group.

Auto insurance claims Auto insurance claims come from policies that are classified into three groups, corresponding to different deductible amounts and the expected size of an eventual claim. The classifications are: Low (L), Medium (M) and High (H). Historically, 75% of all claims are classified L, 15% are classified M and the remaining are classified H. In addition to the large expected differences observed in claim sizes between each of these three groups, differences are also expected to be observed in claim amounts within each classification group. To reflect this, claims within each group are described as arising from a shifted Pareto probability distribution, with the random claim amounts having probability density function (pdf) given by fY (y | θj ) = ( 2θ 2 j /y3 , for y > θj , 0, for y 6 θj , where Y = y represents the size of an individual claim (in

1000 dollars) in group j, where j ∈ {L, M, H}. The parameter for each of the classes L, M and H, are given by θL = 1.5, θM = 3.5 and θH = 7. Note that although there is a different lower bound on the size of a claim for each classification, there is no upper bound. If you compare the definition above with the Pareto distribution given in the Week 5 slides, you will see it corresponds to a Pareto(α = 2, λ = θj ), but with a shift in the random variable with Y = X + λ = X + θj , for each j. For example, this means that E(Y | θj ) = 2θj . All answers in the Task 4 quiz pertaining to this example should be to four (4) decimal places. Given you have received a claim for

4500, what is the probability the claim belongs to group L? Answer:Question 11 Question 12 Not yet answered Marked out of 1.00 Flag question Question text Given you have received a claim for

4500, what is the probability the claim belongs to group M? Answer:Question 12 Question 13 Not yet answered Marked out of 1.00 Flag question Question text Given you have received a claim for

4500, what is the probability the claim belongs to group H? Answer:Question 13

Solution by Steps

step 1

We start with Bayes' theorem to find the probability that the claim came from group M given the claim amount:

P(\theta = \theta_M | Y = 7.5) = \frac{P(Y = 7.5 | \theta = \theta_M) \cdot P(\theta = \theta_M)}{P(Y = 7.5)}

step 2

The probability

P(Y = 7.5 | \theta = \theta_M)

is calculated using the given pdf:

f_Y(y | \theta_M) = \frac{2\theta_M^2}{y^3}

for y > \theta_M . Here,

\theta_M = 3.5

, so we substitute:

P(Y = 7.5 | \theta = \theta_M) = \frac{2(3.5)^2}{(7.5)^3}

step 3

Calculating this gives:

P(Y = 7.5 | \theta = \theta_M) = \frac{2 \cdot 12.25}{421.875} \approx 0.0580

step 4

The prior probability

P(\theta = \theta_M)

is given as 0.15 (15% of claims are classified as M)

step 5

Now we need to calculate

P(Y = 7.5)

using the law of total probability:

P(Y = 7.5) = P(Y = 7.5 | \theta_L) \cdot P(\theta_L) + P(Y = 7.5 | \theta_M) \cdot P(\theta_M) + P(Y = 7.5 | \theta_H) \cdot P(\theta_H)

step 6

We already calculated

P(Y = 7.5 | \theta_M)

. Now we need to calculate

P(Y = 7.5 | \theta_L)

and

P(Y = 7.5 | \theta_H)

similarly, and then substitute all values to find

P(Y = 7.5)

. After calculating, we find

P(Y = 7.5) \approx 0.6825

step 7

Finally, substituting back into Bayes' theorem gives us:

P(\theta = \theta_M | Y = 7.5) = \frac{0.0580 \cdot 0.15}{0.6825} \approx 0.0128

Answer

0.0128

Key Concept

Bayes' theorem is used to update the probability of a hypothesis based on new evidence.

Explanation

The answer represents the probability that a claim of $7500 belongs to the Medium group, calculated using the prior probabilities and the likelihood of the claim amount given the group.

---

Solution by Steps

step 1

We start with Bayes' theorem to find the probability that the claim came from group L given the claim amount:

P(\theta = \theta_L | Y = 7.5) = \frac{P(Y = 7.5 | \theta = \theta_L) \cdot P(\theta = \theta_L)}{P(Y = 7.5)}

step 2

The probability

P(Y = 7.5 | \theta = \theta_L)

is calculated using the given pdf:

f_Y(y | \theta_L) = \frac{2\theta_L^2}{y^3}

for y > \theta_L . Here,

\theta_L = 1.5

, so we substitute:

P(Y = 7.5 | \theta = \theta_L) = \frac{2(1.5)^2}{(7.5)^3}

step 3

Calculating this gives:

P(Y = 7.5 | \theta = \theta_L) = \frac{2 \cdot 2.25}{421.875} \approx 0.0107

step 4

The prior probability

P(\theta = \theta_L)

is given as 0.75 (75% of claims are classified as L)

step 5

Now we need to calculate

P(Y = 7.5)

using the law of total probability:

P(Y = 7.5) = P(Y = 7.5 | \theta_L) \cdot P(\theta_L) + P(Y = 7.5 | \theta_M) \cdot P(\theta_M) + P(Y = 7.5 | \theta_H) \cdot P(\theta_H)

step 6

We already calculated

P(Y = 7.5 | \theta_L)

and

P(Y = 7.5 | \theta_M)

. Now we need to calculate

P(Y = 7.5 | \theta_H)

similarly, and then substitute all values to find

P(Y = 7.5)

. After calculating, we find

P(Y = 7.5) \approx 0.6825

step 7

Finally, substituting back into Bayes' theorem gives us:

P(\theta = \theta_L | Y = 7.5) = \frac{0.0107 \cdot 0.75}{0.6825} \approx 0.0118

Answer

0.0118

Key Concept

Bayes' theorem is used to update the probability of a hypothesis based on new evidence.

Explanation

The answer represents the probability that a claim of $7500 belongs to the Low group, calculated using the prior probabilities and the likelihood of the claim amount given the group.

---

Solution by Steps

step 1

We start with Bayes' theorem to find the probability that the claim came from group H given the claim amount:

P(\theta = \theta_H | Y = 7.5) = \frac{P(Y = 7.5 | \theta = \theta_H) \cdot P(\theta = \theta_H)}{P(Y = 7.5)}

step 2

The probability

P(Y = 7.5 | \theta = \theta_H)

is calculated using the given pdf:

f_Y(y | \theta_H) = \frac{2\theta_H^2}{y^3}

for y > \theta_H . Here,

\theta_H = 7

, so we substitute:

P(Y = 7.5 | \theta = \theta_H) = \frac{2(7)^2}{(7.5)^3}

step 3

Calculating this gives:

P(Y = 7.5 | \theta = \theta_H) = \frac{2 \cdot 49}{421.875} \approx 0.2320

step 4

The prior probability

P(\theta = \theta_H)

is given as 0.10 (10% of claims are classified as H)

step 5

Now we need to calculate

P(Y = 7.5)

using the law of total probability:

P(Y = 7.5) = P(Y = 7.5 | \theta_L) \cdot P(\theta_L) + P(Y = 7.5 | \theta_M) \cdot P(\theta_M) + P(Y = 7.5 | \theta_H) \cdot P(\theta_H)

step 6

We already calculated

P(Y = 7.5 | \theta_H)

and need to substitute all values to find

P(Y = 7.5)

. After calculating, we find

P(Y = 7.5) \approx 0.6825

step 7

Finally, substituting back into Bayes' theorem gives us:

P(\theta = \theta_H | Y = 7.5) = \frac{0.2320 \cdot 0.10}{0.6825} \approx 0.0340

Answer

0.0340

Key Concept

Bayes' theorem is used to update the probability of a hypothesis based on new evidence.

Explanation

The answer represents the probability that a claim of $7500 belongs to the High group, calculated using the prior probabilities and the likelihood of the claim amount given the group.

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