Tag Archives: Conditional Mean

Exam P Practice Problem 93 – Determining Average Claim Frequency

Problem 93-A

An actuary performs a claim frequency study on a group of auto insurance policies. She finds that the probability function of the number of claims per week arising from this set of policies is P(N=n) where n=1,2,3,\cdots. Furthermore, she finds that P(N=n) is proportional to the following function:

    \displaystyle \frac{e^{-2.9} \cdot 2.9^n}{n!} \ \ \ \ \ \ \ n=1,2,3,\cdots

What is the weekly average number of claims arising from this group of insurance policies?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \   2.900

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \   3.015

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \   3.036

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \   3.069

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \   3.195

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Problem 93-B

Let N be the number of taxis arriving at an airport terminal per minute. It is observed that there are at least 2 arrivals of taxis in each minute. Based on a study performed by a traffic engineer, the probability P(N=n) is proportional to the following function:

    \displaystyle \frac{e^{-2.9} \cdot 2.9^n}{n!} \ \ \ \ \ \ \ n=2,3,4,\cdots

What is the average number of taxis arriving at this airport terminal per minute?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \   2.740

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \   2.900

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \   3.339

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \   3.489

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \   3.692

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\copyright \ 2015 \ \ \text{ Dan Ma}

Exam P Practice Problem 68 – Large Claim Studies

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?

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

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

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

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

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

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Problem 68-B

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{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?

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      \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 43 – Joint Random Losses

Problem 43-A

Two random losses X and Y are jointly distributed according to the following density function:

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

Suppose that these two random losses had occurred. If the total loss is 5, what is the expected value of the loss X?

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Problem 43-B

Two random losses X and Y are jointly distributed according to the following density function:

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

Suppose that these two random losses had occurred. If the total loss is 6, what is the expected value of the loss X?

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Exam P Practice Problem 41 – Conditional Expected Number of Balls

Problem 41-A

An insurer has a block of business where the number of claims in a year for a policyholder in the block has the following probability distribution.

            \displaystyle \begin{bmatrix} \text{Number of Claims}&\text{ }&\text{Probability}  \\\text{ }&\text{ }&\text{ } \\ 0&\text{ }&\displaystyle 0.08 \\\text{ }&\text{ }&\text{ }  \\ 1&\text{ }&\displaystyle 0.35 \\\text{ }&\text{ }&\text{ }  \\ 2&\text{ }&\displaystyle 0.40 \\\text{ }&\text{ }&\text{ }  \\ 3&\text{ }&\displaystyle 0.12 \\\text{ }&\text{ }&\text{ }  \\ 4&\text{ }&\displaystyle 0.0375 \\\text{ }&\text{ }&\text{ }  \\ 5&\text{ }&\displaystyle 0.0125   \end{bmatrix}

Two policyholders from this block are randomly selected and observed for a year. It is found that there are exactly four claims for these two policyholders in the past year. What is the probability that one of the policyholders has exactly three claims?

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Problem 41-B

There are two bowls containing red balls and white balls. Bowl 1 contains 5 red balls and 5 white balls. Five balls are randomly selected from Bowl 1 without replacement. Bowl 2 also contains 5 red balls and 5 white balls. Five balls are randomly selected from Bowl 2 with replacement.

If the total number of red balls drawn from the two bowls is eight, what is the expected number of red balls from Bowl 2?

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Exam P Practice Problem 26 – Uniform Distribution

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This post has no alternate problem. It has one problem with 3 parts.

Problem 26

For an office worker, the length of time, X, of the bus ride from home to office follows a uniform distribution from 0 to 20 minutes and the length of time, Y, of the bus ride from office back to home follows a uniform distribution from 0 to 25 minutes.

Suppose that length of bus rides in one direction is independent of the length of bus rides in the other direction.

In recent weeks, the worker finds that the total daily time spent on bus rides always exceeds 20 minutes.

  1. What is the probability that the bus ride to return home will take more than 10 minutes?
  2. What is the expected length of the bus rides from office to home?
  3. What is the variance of the length of the bus rides from office to home?

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Exam P Practice Problem 25 – Uniform Distribution

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This post has no alternate problem. It has one problem with 3 parts.

Problem 25

A real estate property owner is at risk for losses due to two different perils affecting her property. There are no other potential losses in addition to these two different types of losses. Let X be the total amount of the losses in a year due to one peril and let Y be the total amount of the losses in a year due to the other peril. Suppose that X and Y are independent and are identically and uniformly distributed from 0 to 10.

In recent years, the owner finds that the total annual losses (due to both perils) always exceed 10.

  1. What is the probability that the loss X exceeds 5?
  2. What is the expected annual loss X?
  3. What is the variance of the annual loss X?

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Exam P Practice Problem 22 – Poisson Number of Claims

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Problem 22A
Ten percent of the policyholders for an auto insurance company are considered high risk and ninety percent of its policyholders are considered low risk. The number of claims made by a policyholder in a calendar year follows a Poisson distribution with mean \lambda.

For high risk policyholders, \lambda=0.9. For low risk policyholders, \lambda=0.1. An actuary selects one policyholder at random. What is the expected number of claims made by this policyholder in the next calendar year?

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Problem 22B
Use the same information as in Problem 22A. What is the variance of the number of claims made by this randomly selected policyholder?

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