# Exam P Practice Problem 91 – Reviewing a Group of Policyholders

Problem 91-A

A life insurance actuary reviewed a group of policyholders whose policies or contracts were inforce as of last year. The actuary found that 12% of the policyholders who had only a life insurance policy did not survive to this year and that 7.5% of the policyholders who had only an annuity contract did not survive to this year. The actuary also found that 5.9% of the policyholders who had both a life insurance policy and an annuity contract did not survive to this year.

In this group of policyholders, 65% of the policyholders had a life insurance policy and 57% of the policyholders had an annuity contract. Furthermore, each policyholder in this group either had a life insurance policy or an annuity contract.

What is the percentage of the policyholders that did not survive to this year?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 7.8 \%$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 9.0 \%$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 12.0 \%$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 13.4 \%$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 25.4 \%$

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Problem 91-B

A sport coach in a university tracks a group of athletes. The coach finds that 36% of the athletes who play soccer only are first year university students and that 20% of the athletes who are involved only in track and field are first year university students. The coach also discovers that 27% of the athletes participates in both soccer and track and field are first year university students.

According to university records, 45% of the athletes in this group play soccer and 68% of the athletes in this group participate in track and field. Each of the athletes in this group either plays soccer or participates in track and field.

Out of this group of athletes, what is the percentage of the athletes that are not first year university students?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 64 \%$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 67 \%$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 70 \%$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 74 \%$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 80 \%$

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

# Exam P Practice Problem 90 – Insurance Benefits

Problem 90-A

A random loss follows an exponential distribution with mean 20. An insurance reimburses this random loss up to a benefit limit of 30.

When a loss occurs, what is the expected value of the benefit not paid by this insurance policy?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 4.5$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 5.1$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 6.3$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 8.5$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 11.2$

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Problem 90-B

A random loss follows an exponential distribution with mean 100. An insurance reimburses this random loss up to a benefit limit of 200.

When a loss occurs, what is the expected value of the benefit not paid by this insurance policy?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 12.6$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 13.5$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 24.6$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 40.6$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 40.7$

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

# Exam P Practice Problem 89 – Finding Median

Problem 89-A

The random variables $X$ and $Y$ have the following joint density function.

$\displaystyle f(x,y)=\frac{1}{32} \ (4-x) \ \ \ \ \ \ \ 0

Suppose that $m$ is the median of $X+Y$. Which of the following is true about $m$?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 2.5

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 2

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 3

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 3.5

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 3.5

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Problem 89-B

The random variable $X$ has the following density function.

$\displaystyle f(x)=\frac{3}{16000} \ (400-x^2) \ \ \ \ \ \ \ 0

Suppose that $m$ is the median of $X$. Which of the following is true about $m$?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 6

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

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 5.5

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 7

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 7

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

# 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 87 – Modeling Insurance Payments

Problem 87-A

A business owner is facing a risk whose economic loss is modeled by the random variable $X$. The following is the density function of $X$.

$\displaystyle f(x)=\frac{1}{8} \ (4-x) \ \ \ \ \ \ \ \ 0

The business owner purchases an insurance policy to cover this potential loss. The insurance policy pays the business owner 80% of the amount of each loss.

Given that a loss has occurred, what is the probability that the amount of the insurance payment to the business owner is less than 2?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.25$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.36$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.64$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.75$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.86$

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Problem 87-B

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

$\displaystyle f(x)=\frac{1}{1800} \ x \ \ \ \ \ \ \ \ 0

The insurance policy reimburses the policy owner 50% of each loss. Given that a loss has occurred, what is the median amount of the insurance payment made to the policy owner?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 15.00$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 18.65$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 21.21$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 23.63$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 42.43$

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