Tag Archives: Joint Distribution

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<x<4, \ 0<y<x

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<x<4, \ 0<y<4

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<x<\infty, \ \ \ 0<y<\infty

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<x<\infty, \ \ \ 0<y<5

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 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<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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Exam P Practice Problem 46 – Finding Moment of a Sum

Problem 46-A

Suppose that X and Y are random losses that are jointly distributed with the following density function:

      \displaystyle f(x,y)=\frac{1}{40} \ (4-x) \ \ \ \ \ \ 0<x<4, \ \ 0<y<5

Find the second moment of the sum of the two losses.

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Problem 46-B

Suppose that X and Y are random losses that are jointly distributed with the following density function:

      \displaystyle f(x,y)=32 \ x^2 \ y \ e^{-2 \ (x \ + \ y)} \ \ \ \ \ \ 0<x<\infty, \ \ 0<y<\infty

Find the second moment of the sum of the two losses.

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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 38 – Poisson Distribution

Problem 38-A

Two customers (Customer #1 and Customer #2) just purchased identical insurance coverage. The number of claims for each insured is assumed to follow a Poisson distribution with mean 1.5 per year. Assume that the number of claims for Customer #1 is independent of the number of claims for Customer #2.

What is the probability that in the coming year, Customer #1 will have exactly one claim and Customer #2 will have exactly two claims?

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Problem 38-B

The number of customers visiting a jewelry store on a weekday has a Poisson distribution with mean 4 per hour. Assume that for this jewelry store the number of customers in any given hour on a weekday is independent of the number of customers in any other hour on a weekday.

A prospective buyer of this jewelry store observes the business on a Wednesday for two one-hour periods (from 1 PM to 2 PM and 4 to 5 PM).

What is the probability that there will be 3 customers visiting from 1 PM to 2 PM and 5 customers visiting from 4 to 5 PM?

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Exam P Practice Problem 36 – Number of Claims

This post has no alternate problem. It has one problem with 2 parts.

Problem 36

A claim examiner of an insurer reviews the claim history of two independent insureds. Let X be the annual number of claims of the first insured (Insured # 1). Let Y be the annual number of claims of the second insured (Insured # 2). The claim examiner finds tht X follows a Poisson distribution with mean 1, and that Y follows a distribution with the following probability function.

      \displaystyle P(Y=y)=\frac{3!}{y! (3-y)!} \ \biggl(\frac{1}{3}\biggr)^y \ \biggl(\frac{2}{3}\biggr)^{3-y} \ \ \ \ \ \ y=0,1,2,3
  1. Between these two insureds, what is the probability that one of the insureds has two more claims than the other insured in a year?
  2. Given that one of the insured has two more claims than the other insured, what is the probability that Insured # 1 has more claims than Insured # 2?

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