# 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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# 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

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

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

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. An insurance policy is purchased to reimburse the loss up to a maximum limit of $m$ where $0.

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

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

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

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

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