Category Archives: Insurance and Risk Management

Exam P Practice Problem 68 – Large Claim Studies

By Dan Ma on April 13, 2013 | Leave a comment

Problem 68-A

An insurance company has a large block of auto insurance policies such that the claim sizes (in thousands) from a policy in this block are modeled by the random variable $X$ . The following is the probability density function of $X$ .

$\displaystyle f(x)=\frac{3}{16000} \ (400-x^2) \ \ \ \ \ \ \ 0<x<20$

An actuary is hired to study the large claims arising from these auto insurance policies. In this study, any claim size over ten thousands is considered a large claim.

What is the mean size of the claims studied by this actuary?

$\text{ }$

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7,500$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 13,500$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 14,219$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 15,000$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 17,500$

$\text{ }$

Problem 68-B

$\displaystyle f(x)=\frac{625}{312 \ x^3} \ \ \ \ \ \ \ 1<x<25$

An actuary is hired to study the large claims arising from these auto insurance policies. In this study, any claim size over five thousands is considered a large claim.

What is the mean size of the claims studied by this actuary?

$\text{ }$

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1,923$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6,923$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 5,321$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8,333$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 15,000$

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Exam P Practice Problem 67 – Statistical Studies of Insured Drivers

By Dan Ma on April 12, 2013 | Leave a comment

Problem 67-A

An auto insurance company performed a statistical study on its insured drivers. The following table shows the results.

$\displaystyle \begin{bmatrix} \text{Age Group}&\text{ }&\text{Percentage}&\text{ }&\text{Annual Probability of} \\ \text{ }&\text{ }&\text{of its Drivers}&\text{ }&\text{At Least One Claim} \\\text{ }&\text{ }&\text{ } \\ \text{16-20}&\text{ }&15 \% &\text{ }&0.18 \\\text{ }&\text{ }&\text{ } \\ \text{21-30}&\text{ }&20 \% &\text{ }&0.12 \\\text{ }&\text{ }&\text{ } \\ \text{31-50}&\text{ } &30 \% &\text{ }&0.08 \\\text{ }&\text{ }&\text{ } \\ \text{51-70}&\text{ }&25 \% &\text{ }&0.09 \\\text{ }&\text{ }&\text{ } \\ \text{71 and up}&\text{ }&10 \% &\text{ }&0.11 \end{bmatrix}$

The authors of the statistical study also found that for any insured driver in the study, the annual number of claims follows a Poisson distribution.

Suppose that an insured driver in the study had exactly 2 claims in the past year. What is the probability that the insured driver is from the age group 16-20?

$\text{ }$

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.150$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.223$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.249$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.376$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.415$

$\text{ }$

Problem 67-B

An auto insurance company performed a statistical study on its younger insured drivers (under 35 years of age). The following table shows the results.

$\displaystyle \begin{bmatrix} \text{Age Group}&\text{ }&\text{Percentage}&\text{ }&\text{Annual Probability of} \\ \text{ }&\text{ }&\text{of its Drivers}&\text{ }&\text{At Least One Claim} \\\text{ }&\text{ }&\text{ } \\ \text{16-17}&\text{ }&12 \% &\text{ }&0.18 \\\text{ }&\text{ }&\text{ } \\ \text{18-24}&\text{ }&38 \% &\text{ }&0.10 \\\text{ }&\text{ }&\text{ } \\ \text{25-34}&\text{ } &50 \% &\text{ }&0.06 \end{bmatrix}$

The authors of the statistical study also found that for any insured driver in the study, the annual number of claims follows a Poisson distribution. Furthermore, for any insured driver in the study, the number of claims in one year is independent of the number of claims in any other year.

Suppose that in a 2-year period, an insured driver in the study had exactly 1 claim in year 1 and exactly 2 claims in year 2. What is the probability that the insured driver is from the age group 16-17?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.120$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.229$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.241$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.329$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.576$

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Answers

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$\copyright \ 2013$

Posted in: Insurance and Risk Management, Poisson distribution, Practice Problems | Tagged: Actuarial Exam, Bayes' Theorem, CAS Exam 1, Exam P, Exam P Practice Problems, Insurance and risk management, Poisson distribution, Probability, SOA Exam P

Exam P Practice Problem 65 – Total Insurance Payment

By Dan Ma on April 9, 2013 | Leave a comment

Problem 65-A

The number of random losses in a calendar year for an individual has a Poisson distribution with mean 1. When a loss occurs, the individual loss amount is either 2 or 4, with probabilities 0.6 and 0.4, respectively.

When multiple losses occur for this individual, the individual loss amounts are independent.

This individual purchases an insurance policy to cover the random losses with a deductible of 1 per loss.

In the next calendar year, let $S$ be the total payment made by the insurance company to the insured. Calculate $P(2 \le S \le 4)$ .

$\text{ }$

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.12204$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.16974$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.29380$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.31689$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.34277$

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Problem 65-B

The number of claims in a calendar year for an insured has a Poisson distribution with mean 1.2. When a claim occurs, the individual claim amount is either 10 or 20, with probabilities 0.8 and 0.2, respectively.

When multiple claims occur for this insured, the individual claim amounts are independent.

This individual purchases an insurance policy to cover the random losses with a deductible of 5 per loss.

In the next calendar year, let $S$ be the total payment made by the insurance company to the insured. Calculate $P(10 \le S < 30)$ .

$\text{ }$

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1010$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.2986$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3709$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3826$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3906$

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Answers

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$\copyright \ 2013$

Posted in: Insurance and Risk Management, Poisson distribution, Practice Problems | Tagged: Actuarial Exam, Binomial distribution, CAS Exam 1, Exam P, Exam P Practice Problems, Insurance and risk management, Poisson distribution, Probability, SOA Exam P

Exam P Practice Problem 61 – Claim Size of Auto Insurance Policies

By Dan Ma on March 31, 2013 | 1 Comment

Problem 61-A

An insurance company has a block of auto insurance policies. The claim size (in thousands) for a policy in this block of auto insurance policies is modeled by the random variable $Y=X^2$ where $X$ has an exponential distribution with mean 1.25.

What is the expected claim size for such an auto insurance policy?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ 1250$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ 1563$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ 2500$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ 2755$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ 3125$

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Problem 61-B

What is the standard deviation of the claim size for such an auto insurance policy?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ 1600$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ 5120$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ 9756.43$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ 11448.67$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ 12541.39$

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Answers

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Posted in: Exponential distribution, Insurance and Risk Management, Practice Problems | Tagged: Actuarial Exam, CAS Exam 1, Exam P, Exam P Practice Problems, Expected Value, Exponential distribution, Insurance and risk management, Probability, SOA Exam P

Exam P Practice Problem 60 – Health Insurance Claim Frequency

By Dan Ma on March 29, 2013 | Leave a comment

Problem 60-A

An insurance company issued health insurance policies to individuals. The company determined that $Y$ , the number of claims filed by an insured in a year, is a random variable with the following probability function.

$\displaystyle P(Y=y)=0.45 \ (0.55)^{\displaystyle y} \ \ \ \ \ \ y=0,1,2,3,\cdots$

What is the probability that a random selected insured from this group of insured individuals will file more than 5 claims in a year?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0226$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0277$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0357$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0503$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0749$

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Problem 60-B

$\displaystyle P(Y=y)=0.45 \ (0.55)^{\displaystyle y} \ \ \ \ \ \ y=0,1,2,3,\cdots$

The number of claims filed by one insured individual is independent of the number of claims filed by any other insured individual.

An actuary studied three randomly selected insured individuals from this group of individuals who purchased health policies from this company. What is the probability that these three insured individuals will file more than 6 claims in a year?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0457$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0706$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1495$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.2201$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.2406$

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Posted in: Independent Sum, Insurance and Risk Management, Negative Binomial Distribution, Practice Problems | Tagged: Actuarial Exam, CAS Exam 1, Exam P, Exam P Practice Problems, Geometric distribution, Independent sum, Insurance and risk management, Negative Binomial Distribution, Probability, SOA Exam P

Exam P Practice Problem 58 – Dental Care and Vision Care Expenses

By Dan Ma on March 28, 2013 | Leave a comment

Problem 58-A

A health plan offers dental care and vision care benefits. Let $X$ represents the total annual amount (in millions) paid in dental care benefits. Let $Y$ represents the total annual amount (in millions) paid in vision care benefits.

The health plan determined that

$X=K^2$ where $K$ follows a normal distribution with mean 0 and variance 1,
$Y=L^2$ where $L$ follows a normal distribution with mean 0 and variance 2, and
$K$ and $L$ are independent.

Given that the total annual vision care benefits paid by the health plan exceeds 2.5 millions, what is the probability that the total annual dental care benefits paid by the health plan exceeds 2 millions?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0228$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0793$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1586$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ 0.8416$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ 0.9207$

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Problem 58-B

The health plan determined that

$X=2.5 K^2$ where $K$ follows a normal distribution with mean 0 and variance 1,
$Y=5 L^2$ where $L$ follows a normal distribution with mean 0 and variance 1, and
$K$ and $L$ are independent.

What is the probability that the total annual dental care benefits exceeds 3 millions and that the total annual vision care benefits exceeds 4 millions?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1013$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ 0.4565$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ 0.6266$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ 0.7286$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ 0.7881$

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Posted in: Insurance and Risk Management, Normal Distribution, Practice Problems | Tagged: Actuarial Exam, CAS Exam 1, Exam P, Exam P Practice Problems, Independent Random Varaibles, Insurance and risk management, Joint Distribution, Normal Distribution, Probability, SOA Exam P, Variance

Exam P Practice Problem 56 – Reporting of Auto Accidents

By Dan Ma on March 21, 2013 | Leave a comment

Problem 56-A

An insurer sells auto insurance policies that provide collision coverage to drivers. The collision accidents reported by drivers are uniformly distributed across the days of the week.

The day of reporting an accident is independent of the day of reporting of any other accident.

Suppose that in one week, 10 collision accidents are reported to the insurer. What is the probability that more than 3 accidents are reported on Saturday and Sunday?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1269$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3127$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.4218$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.5782$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.6873$

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Problem 56-B

An insurer sells auto insurance policies that provide collision coverage to drivers. The collision accidents reported by drivers are uniformly distributed across the days of the week.

The day of reporting an accident is independent of the day of reporting of any other accident. The number of accidents reported in one week is also independent of the number of accidents reported in any other week.

Suppose that in one week, 10 collision accidents are reported to the insurer and in the following week, 12 collision accidents are reported to the insurer. What is the probability that more than 20% of the accidents from these two weeks are reported on Saturday and Sunday?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0571$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0886$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.2028$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.7972$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.9114$

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Posted in: Insurance and Risk Management, Practice Problems | Tagged: Actuarial Exam, Binomial distribution, Exam P, Exam P Practice Problems, Insurance and risk management, Probability

Exam P Practice Problem 55 – Expected Benefit Payment

By Dan Ma on March 20, 2013 | Leave a comment

Problem 55-A

The following is the joint density function of two random losses $X$ and $Y$ .

$\displaystyle f(x,y)=\frac{3}{16} \ x^2 \ \ \ \ \ \ \ \ \ 0<x<2, \ 0<y<2$

An insurance policy is purchased to cover the total loss $X+Y$ subject to a deductible of 2.

When the losses $X$ and $Y$ occur, what is the expected benefit paid by this insurance policy?

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

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 0.60$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 0.78$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 1.86$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 2.50$

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Problem 55-B

The following is the joint density function of two random losses $X$ and $Y$ .

$\displaystyle f(x,y)=\frac{1}{64} \ x \ y \ \ \ \ \ \ \ \ \ 0<x<4, \ 0<y<4$

An insurance policy is purchased to cover the total loss $X+Y$ subject to a deductible of 4.

When the losses $X$ and $Y$ occur, what is the expected benefit paid by this insurance policy?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 5.333$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 1.833$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 1.333$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 1.467$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 1.296$

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Posted in: Insurance and Risk Management, Practice Problems | Tagged: Actuarial Exam, Exam P, Exam P Practice Problems, Expected Value, Insurance and risk management, Joint Distribution, Probability

Exam P Practice Problem 54 – Expected Insurance Payment

By Dan Ma on March 19, 2013 | Leave a comment

Problem 54-A

An insurance policy is purchased to reimburse a loss that is modeled by the following probability density function:

$\displaystyle f(x)=\frac{30}{1024} \ x^2 \ (4-x)^2 \ \ \ \ \ \ \ 0<x<4$

This insurance policy has a deductible of 1 with an additional provision that any loss that exceeds the deductible will be paid in full to the policyholder.

When there is a loss, what is the expected amount paid to the policyholder under this policy?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 1.028$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 1.598$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 1.836$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 1.925$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 2.000$

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Problem 54-B

An insurance policy is purchased to reimburse a loss that is modeled by the following probability density function:

$\displaystyle f(x)=\frac{5}{256} \ x^3 \ (4-x) \ \ \ \ \ \ \ 0<x<4$

This insurance policy has a deductible of 2 with an additional provision that any loss that exceeds the deductible will be paid in full to the policyholder.

When there is a loss, what is the expected amount paid to the policyholder under this policy?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.750$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 2.375$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 2.667$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 3.375$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 3.667$

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Answers

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Posted in: Insurance and Risk Management, Practice Problems | Tagged: Actuarial Exam, Exam P, Exam P Practice Problems, Expected Value, Insurance and risk management, Probability

Exam P Practice Problem 53 – Hospital Expense Plans

By Dan Ma on March 18, 2013 | Leave a comment

Problem 53-A

An insurer sells a hospital expense plan that pays a fixed sum per day of hospitalization. Suppose that the number of days of hospitalization in a year for someone insured under this plan has a Poisson distribution with mean 0.8.

In each calendar year, the plan pays 2,000 for each day of hospitalization subject to the condition that the first two days of hospitalization are the responsibilities of the insured.

What is the expected payment for hospitalization during a calendar year under this hospital expense plan?

$\displaystyle (A) \ \ \ \ \ \ \ 116.24$

$\displaystyle (B) \ \ \ \ \ \ \ 244.75$

$\displaystyle (C) \ \ \ \ \ \ \ 305.93$

$\displaystyle (D) \ \ \ \ \ \ \ 785.26$

$\displaystyle (E) \ \ \ \ \ \ \ 1600.00$

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Problem 53-B

$\displaystyle P(X=x)=\frac{3}{4^{x+1}} \ \ \ \ \ \ \ \ x=0,1,2,3,\cdots$

In each calendar year, the plan pays 1,000 for each day of hospitalization subject to the condition that the first day of hospitalization is the responsibility of the insured.

What is the expected payment for hospitalization during a calendar year under this hospital expense plan?

$\displaystyle (A) \ \ \ \ \ \ \ 76.83$

$\displaystyle (B) \ \ \ \ \ \ \ 83.33$

$\displaystyle (C) \ \ \ \ \ \ \ 111.11$

$\displaystyle (D) \ \ \ \ \ \ \ 145.83$

$\displaystyle (E) \ \ \ \ \ \ \ 333.33$

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Posted in: Insurance and Risk Management, Poisson distribution, Practice Problems | Tagged: Actuarial Exam, Exam P, Exam P Practice Problems, Expected Value, Geometric distribution, Insurance and risk management, Poisson distribution, Probability

SOA Exam P / CAS Exam 1

Category Archives: Insurance and Risk Management

Exam P Practice Problem 68 – Large Claim Studies

Exam P Practice Problem 67 – Statistical Studies of Insured Drivers

Exam P Practice Problem 65 – Total Insurance Payment

Exam P Practice Problem 61 – Claim Size of Auto Insurance Policies

Exam P Practice Problem 60 – Health Insurance Claim Frequency

Exam P Practice Problem 58 – Dental Care and Vision Care Expenses

Exam P Practice Problem 56 – Reporting of Auto Accidents

Exam P Practice Problem 55 – Expected Benefit Payment

Exam P Practice Problem 54 – Expected Insurance Payment

Exam P Practice Problem 53 – Hospital Expense Plans

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