Tag Archives: Expected Value

Exam P Practice Problem 102 – estimating claim costs

Problem 102-A

Insurance claims modeled by a distribution with the following cumulative distribution function.

    \displaystyle  F(x) = \left\{ \begin{array}{ll}           \displaystyle  0 &\ \ \ \ \ \ x \le 0 \\            \text{ } & \text{ } \\          \displaystyle  \frac{1}{1536} \ x^4 &\ \ \ \ \ \ 0 < x \le 4 \\           \text{ } & \text{ } \\          \displaystyle  1-\frac{2}{3} x+\frac{1}{8} x^2- \frac{1}{1536} \ x^4 &\ \ \ \ \ \ 4 < x \le 8 \\           \text{ } & \text{ } \\          \displaystyle  1 &\ \ \ \ \ \ x > 8 \\                      \end{array} \right.

The insurance company is performing a study on all claims that exceed 3. Determine the mean of all claims being studied.

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 4.8

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 4.9

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 5.0

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 5.1

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 5.2

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Problem 102-B

Insurance claims modeled by a distribution with the following cumulative distribution function.

    \displaystyle  F(x) = \left\{ \begin{array}{ll}           \displaystyle  0 &\ \ \ \ \ \ x \le 0 \\            \text{ } & \text{ } \\          \displaystyle  \frac{1}{50} \ x^2 &\ \ \ \ \ \ 0 < x \le 5 \\           \text{ } & \text{ } \\          \displaystyle  -\frac{1}{50} x^2+\frac{2}{5} x- 1 &\ \ \ \ \ \ 5 < x \le 10 \\           \text{ } & \text{ } \\          \displaystyle  1 &\ \ \ \ \ \ x > 10 \\                      \end{array} \right.

The insurance company is performing a study on all claims that exceed 4. Determine the mean of all claims being studied.

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 5.9

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 6.0

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 6.1

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 6.2

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 6.3

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Answers can be found in this page.

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Exam P Practice Problem 97 – Variance of Claim Sizes

Problem 97-A

For a type of insurance policies, the following is the probability that the size of claim is greater than x.

    \displaystyle  P(X>x) = \left\{ \begin{array}{ll}           \displaystyle  1 &\ \ \ \ \ \ x \le 0 \\            \text{ } & \text{ } \\          \displaystyle  \biggl(1-\frac{x}{10} \biggr)^6 &\ \ \ \ \ \ 0<x<10 \\           \text{ } & \text{ } \\           0 &\ \ \ \ \ \ x \ge 10           \end{array} \right.

Calculate the variance of the claim size for this type of insurance policies.

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \frac{10}{7}

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \frac{75}{49}

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \frac{95}{49}

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \frac{15}{7}

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \frac{25}{7}

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Problem 97-B

For a type of insurance policies, the following is the probability that the size of a claim is greater than x.

    \displaystyle  P(X>x) = \left\{ \begin{array}{ll}           \displaystyle  1 &\ \ \ \ \ \ x \le 0 \\            \text{ } & \text{ } \\          \displaystyle  \biggl(\frac{250}{x+250} \biggr)^{2.25} &\ \ \ \ \ \ x>0 \\                             \end{array} \right.

Calculate the expected claim size for this type of insurance policies.

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 200.00

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 203.75

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 207.67

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 217.32

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 232.74

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Exam P Practice Problem 96 – Expected Insurance Payment

Problem 96-A

An insurance policy is purchased to cover a random loss subject to a deductible of 1. The cumulative distribution function of the loss amount X is:

    \displaystyle  F(x) = \left\{ \begin{array}{ll}           \displaystyle  0 &\ \ \ \ \ \ x<0 \\            \text{ } & \text{ } \\          \displaystyle  \frac{3}{25} \ x^2 - \frac{2}{125} \ x^3 &\ \ \ \ \ \ 0 \le x<5 \\           \text{ } & \text{ } \\           1 &\ \ \ \ \ \ 5<x           \end{array} \right.

Given a random loss X, determine the expected payment made under this insurance policy?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 0.50

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 1.54

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 1.72

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 4.63

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 6.26

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Problem 96-B

An insurance policy is purchased to cover a random loss subject to a deductible of 2. The density function of the loss amount X is:

    \displaystyle  f(x) = \left\{ \begin{array}{ll}                     \displaystyle  \frac{3}{8} \biggl(1- \frac{1}{4} \ x + \frac{1}{64} \ x^2 \biggr) &\ \ \ \ \ \ 0<x<8 \\           \text{ } & \text{ } \\           0 &\ \ \ \ \ \ \text{otherwise}           \end{array} \right.

Given a random loss X, what is the expected benefit paid by this insurance policy?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 0.51

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 0.57

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 0.63

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 1.60

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 2.00

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\copyright \ 2016 - \text{Dan Ma}

Exam P Practice Problem 88 – Expected Value of Insurance Payments

Problem 88-A

A random loss X has a uniform distribution over the interval 0<x<20. An insurance policy is purchased to reimburse the loss up to a maximum limit of m where 0<m<20.

The expected value of the benefit payment under this policy is 8.4. Calculate the value of m.

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \   8.7

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \   9.0

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \   12.0

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \   13.6

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \   18.3

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Problem 88-B

An individual purchases an insurance policy to cover a loss X whose density function is:

    \displaystyle f(x)=\frac{2}{25} \ (5-x) \ \ \ \ \ \ \ \ 0<x<5

The insurance policy reimburses the policy owner up to a benefit limit of 4 for each loss. What is the expected value of insurance payment made to the policy owner?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 1.35

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 1.41

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 1.49

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 1.65

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 1.67

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\copyright \ 2014 \ \ \text{ Dan Ma}

Exam P Practice Problem 86 – Finding Mean and Variance

The following is the cumulative distribution function of the random variable X.

    \displaystyle F(x)=\left\{\begin{matrix} \displaystyle 0&\ \ \ \ \ \ x < 0 \\{\text{ }}& \\{\displaystyle \frac{(x+2)^2}{100}}&\ \ \ \ \ \ 0 \le x <6 \\{\text{ }}& \\{\displaystyle 1}&\ \ \ \ \ \ 6 \le x <\infty  \end{matrix}\right.

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Problem 86-A

Calculate the expected value of X.

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \   2.16

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \   3.35

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \   4.32

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \   6.00

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \   6.67

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Problem 86-B

Calculate the variance of X.

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 3.240

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 3.658

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 3.957

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 4.694

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 5.556

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\copyright \ 2014 \ \ \text{ Dan Ma}

Exam P Practice Problem 77 – Estimating Random Claim Sizes

Problem 77-A

The probability distribution of the claim size from an auto insurance policy randomly selected from a large pool of policies is described by the following density function.

    \displaystyle f(x)=\frac{3}{1000} \ (50-5x+\frac{1}{8} \ x^2), \ \ \ \ \ \ \ \ \ \ 0<x<20

What is the probability that a randomly selected claim from this insurance policy is within 120% of the mean claim size?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.50

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.85

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.88

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.91

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.95

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Problem 77-B

The probability distribution of the claim size from an auto insurance policy randomly selected from a large pool of policies is described by the following density function.

    \displaystyle f(x)=\frac{3}{2500} \ (100x-20x^2+ x^3), \ \ \ \ \ \ \ \ \ \ 0<x<10

What is the probability that a randomly selected claim from this insurance policy is within one-half of a standard deviation of the mean claim size?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.34

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.37

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.60

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.62

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.64

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\copyright \ 2013 \ \ \text{Dan Ma}

Exam P Practice Problem 76 – Quantifying Average Random Loss

Both Problem 76-A and Problem 76-B use the following information.

A property owner faces a series of independent random losses. Each loss is either 10 (with probability 0.4) or 50 (with probability 0.6).

Three such random losses are selected.

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Problem 76-A

What is the probability that the mean of the three losses is less than 30?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.06

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.29

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.32

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.35

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.43

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Problem 76-B

What is the expected value of the mean of the three losses?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 30

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 32

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 33

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 34

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 35

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\copyright \ 2013 \ \ \text{Dan Ma}

Exam P Practice Problem 75 – Travel Time to Work By Train

Both Problem 75-A and Problem 75-B use the following information.

A worker travels to work by train 5 days a week (Monday to Friday). The length of a train ride (in minutes) to work follows a continuous uniform distribution from 10 to 40.

The lengths of the train ride across the days of the week are independent.

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Problem 75-A

What is the probability that the shortest train ride during a work week is between 15 and 20 minutes?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.025

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.039

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.045

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.053

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.064

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Problem 75-B

What is the expected value of the longest train ride during a work week?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 25.0

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 25.9

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 28.2

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 33.3

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 35.7

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\copyright \ 2013-2016 \ \ \text{Dan Ma}

Exam P Practice Problem 74 – Review of Auto Insurance Claims

Both Problem 74-A and Problem 74-B use the following information.

An insurer issued policies to cover a large number of automobiles. Claim amounts (in thousands) from these policies are independent and are modeled by a continuous uniform distribution on (0,10).

The insurer randomly selects five claims for review.

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Problem 74-A

What is the probability that the minimum claim amount is between 2 thousands and 6 thousands?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.31

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.32

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.33

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.34

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.75

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Problem 74-B

What is the expected value of the maximum claim amount?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 5.0

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 5.5

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7.6

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8.3

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8.5

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\copyright \ 2013 \ \ \text{Dan Ma}

Exam P Practice Problem 73 – Wait Time at a Busy Restaurant

Both Problem 73-A and Problem 73-B use the following information.

A certain restaurant is very busy in the evening time during the weekend. When customers arrive, they typically have to wait for a table.

When a customer has to wait for a table, the wait time (in minutes) follows a distribution with the following density function.

    \displaystyle f(x)=\frac{1}{1800} \ x, \ \ \ \ \ \ \ \ \ 0<x<60

A customer plans to dine at this restaurant on five Saturday evenings during the next 3 months. Assume that the customer will have to wait for a table on each of these evenings.

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Problem 73-A

What is the probability that the minimum wait time for a table during the next 3 months for this customer will be more than half an hour?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.24

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.25

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.36

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.42

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.75

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Problem 73-B

What is the mean of the maximum wait time (in minutes) for a table during the next 3 months for this customer?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 40.0

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 50.5

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 51.4

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 54.5

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 55.4

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\copyright \ 2013 \ \ \text{Dan Ma}