Tag Archives: Moment Generating Function

Exam P Practice Problem 100 – find the variance of loss in profit

Problem 100-A

The monthly amount of time X (in hours) during which a manufacturing plant is inoperative due to equipment failures or power outage follows approximately a distribution with the following moment generating function.

    \displaystyle M(t)=\biggl( \frac{1}{1-7.5 \ t} \biggr)^2

The amount of loss in profit due to the plant being inoperative is given by Y=12 X + 1.25 X^2.

Determine the variance of the loss in profit.

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \text{279,927.20}

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \text{279,608.20}

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \text{475,693.76}

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \text{583,358.20}

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \text{601,769.56}

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Problem 100-B

The weekly amount of time X (in hours) that a manufacturing plant is down (due to maintenance or repairs) has an exponential distribution with mean 8.5 hours.

The cost of the downtime, due to lost production and maintenance and repair costs, is modeled by Y=15+5 X+1.2 X^2.

Determine the variance of the cost of the downtime.

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \text{130,928.05}

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \text{149,368.45}

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \text{181,622.05}

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \text{188,637.67}

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \text{195,369.15}

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Answers can be found in this page.

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Exam P Practice Problem 49 – Aggregate Claim Costs

Problem 49-A

The aggregate claim amount (in millions) in a year for a block of fire insurance policies is modeled by a random variable Y=e^X where X has a normal distribution with mean 4 and variance 2. What is the expected annual aggregate claim amount?

      \displaystyle (A) \ \ \ \ \ \ \ 403.43

      \displaystyle (B) \ \ \ \ \ \ \ 244.69

      \displaystyle (C) \ \ \ \ \ \ \ 148.41

      \displaystyle (D) \ \ \ \ \ \ \ 90.02

      \displaystyle (E) \ \ \ \ \ \ \ 54.60

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Problem 49-B

The aggregate claim amount (in millions) in a year for a block of fire insurance policies is modeled by a random variable Y=e^X where X has a normal distribution with mean 1.15 and variance 1.2. What is the probability that the annual aggregate claim amount will be less than the expected annual aggregate claim amount?

      \displaystyle (A) \ \ \ \ \ \ \ 0.5000

      \displaystyle (B) \ \ \ \ \ \ \ 0.6915

      \displaystyle (C) \ \ \ \ \ \ \ 0.7088

      \displaystyle (D) \ \ \ \ \ \ \ 0.8749

      \displaystyle (E) \ \ \ \ \ \ \ 0.9599

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Exam P Practice Problem 28 – Wait Time to See Doctor

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Problem 28A

In the previous years, the wait time (in minutes) for a patient to see a doctor at a large medical clinic followed a distribution with the following moment generating function:

    \displaystyle M(t)=\frac{4}{4-100t} \ \ \ \ \ \ \ \ \ \ 4-100t>0

In the current year, the administrator of the clinic still believes that the wait time to see a doctor follows the same distribution except that the mean wait time has increased by 50% (due to the fact that a higher volume of patients is being served by the same number of medical doctors).

What is the probability that in the current year the wait time for a patient to see a doctor is less than 30 minutes?

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Problem 28B
Use the same information as in Problem 28A. What is the percent increase in the variance of the wait time to see a doctor between the current year and the previous years?

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