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

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

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 1.6.

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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# Exam P Practice Problem 60 – Health Insurance Claim Frequency

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

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.

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

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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# Exam P Practice Problem 59 – Joint Distributions

Problem 59-A

Two random losses $X$ and $Y$ are jointly modeled by the following density function:

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

Suppose that both of these losses had occurred. Given that $X$ exceeds 2, what is the probability that $Y$ is less than 2?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.4000$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.4667$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.7518$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.8571$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.9375$

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Problem 59-B

Two random losses $X$ and $Y$ are jointly modeled by the following density function:

$\displaystyle f(x,y)=\frac{1}{96} \ (x+2y) \ \ \ \ \ \ 0

Suppose that both of these losses had occurred. What is the probability that only one of them exceeds 2?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.1250$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.2083$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.2917$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.3750$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.5000$

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

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

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

• $X=2.5 K^2$ 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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# Exam P Practice Problem 57 – Lifetimes of Machines

Problem 57-A

A factory owner purchased two identical machines for her factory. Let $X$ and $Y$ be the lifetimes (in years) of these two machines. These lifetimes are modeled by the following joint probability density function.

$\displaystyle f(x,y)=\frac{0.01}{\sqrt{x} \ \sqrt{y}} \ e^{-0.2 \sqrt{x}} \ e^{-0.2 \sqrt{y}} \ \ \ \ \ \ \ 0

The machine whose lifetime is modeled by the random variable $Y$ came online 2 years after the beginning of operation of the machine that is modeled by the random variable $X$.

Given that $X$ exceeds 2, that is the probability that $Y$ exceeds 3?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 0.2928$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 0.4670$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 0.5330$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 0.7072$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 0.7536$

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Problem 57-B

A company purchased two machines for its factory. Let $X$ and $Y$ be the lifetimes (in years) of these machines. The following is the joint density function of their lifetimes.

$\displaystyle f(x,y)=\frac{3}{125} \ y \ e^{-0.3 x} \ \ \ \ \ \ \ 0

The machine whose lifetime is modeled by the random variable $Y$ came online after the failure of the machine whose lifetime is modeled by $X$.

What is the variance of the total time of operation of these two machines?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 12.50$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 13.60$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 17.20$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 19.85$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 23.61$

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# Exam P Practice Problem 56 – Reporting of Auto Accidents

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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# Exam P Practice Problem 55 – Expected Benefit Payment

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

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

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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# Exam P Practice Problem 54 – Expected Insurance Payment

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

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

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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# Exam P Practice Problem 53 – Hospital Expense Plans

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

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 the following probability function.

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

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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# Exam P Practice Problem 52 – Reliability of Refrigerators

Problem 52-A

The time from initial purchase to the time of the first major repair (in years) for a brand of refrigerators is modeled by the random variable $Y=e^X$ where $X$ is normally distributed with mean 1.2 and variance 2.25.

A customer just bought a brand new refrigerator of this particular brand. The refrigerator came with a two-year warranty. During the warranty period, any repairs, both minor and major, are the responsibilities of the manufacturer.

What is the probability that the newly purchased refrigerator will not require major repairs during the warranty period?

$\displaystyle (A) \ \ \ \ \ \ \ \ 0.2451$

$\displaystyle (B) \ \ \ \ \ \ \ \ 0.2981$

$\displaystyle (C) \ \ \ \ \ \ \ \ 0.3669$

$\displaystyle (D) \ \ \ \ \ \ \ \ 0.6331$

$\displaystyle (E) \ \ \ \ \ \ \ \ 0.7549$

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Problem 52-B

The time from initial purchase to the time of the first major repair (in years) for a brand of refrigerators is modeled by the random variable $Y=e^X$ where $X$ is normally distributed with mean 0.8 and standard deviation 1.5.

What is the median length of time (from initial purchase) that is free of any need for major repairs?

$\displaystyle (A) \ \ \ \ \ \ \ \ 0.80 \text{ years}$

$\displaystyle (B) \ \ \ \ \ \ \ \ 2.23 \text{ years}$

$\displaystyle (C) \ \ \ \ \ \ \ \ 3.50 \text{ years}$

$\displaystyle (D) \ \ \ \ \ \ \ \ 4.71 \text{ years}$

$\displaystyle (E) \ \ \ \ \ \ \ \ 6.86 \text{ years}$

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