## Practice Problems for Conditional Distributions, Part 2

The following are practice problems on conditional distributions. The thought process of how to work with these practice problems can be found in the blog post Conditionals Distribution, Part 2.

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Practice Problems

Practice Problem 1

Suppose that $X$ is the lifetime (in years) of a brand new machine of a certain type. The following is the density function.

$\displaystyle f(x)=\frac{1}{8 \sqrt{x}}, \ \ \ \ \ \ \ \ \ 1

You just purchase a 9-year old machine of this type that is in good working condition. Compute the following:

• What is the expected lifetime of this 9-year old machine?
• What is the expected remaining life of this 9-year old machine?

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Practice Problem 2

Suppose that $X$ is the total amount of damages (in millions of dollars) resulting from the occurrence of a severe wind storm in a certain city. The following is the density function of $X$.

$\displaystyle f(x)=\frac{81}{(x+3)^4}, \ \ \ \ \ \ \ \ \ 0

Suppose that the next storm is expected to cause damages exceeding one million dollars. Compute the following:

• What is the expected total amount of damages for the next storm given that it will exceeds one million dollars?
• The city has a reserve fund of one million dollars to cover the damages from the next storm. Given the amount of damages for the next storm will exceeds one million dollars, what is the expected total amount of damages in excess of the amount in the reserve fund?

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Answers

The thought process of how to work with these practice problems can be found in the blog post Conditionals Distribution, Part 2.

Practice Problem 1

$\displaystyle E(X \lvert X>9)=\frac{49}{3}=16.33 \text{ years}$

$\displaystyle E(X-9 \lvert X>9)=\frac{22}{3}=7.33 \text{ years}$

Practice Problem 2

$\displaystyle E(X \lvert X>1)=3 \text{ millions}$

$\displaystyle E(X-1 \lvert X>1)=2 \text{ millions}$

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$\copyright \ 2013 \text{ by Dan Ma}$

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