# 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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$\copyright \ 2013$