**Problem 9a**

Suppose that when a policyholder incurs a loss, the size of the loss (in thousands of dollars) is where the probability density function of is where . Find the variance of the size of loss.

**Problem 9b**

Suppose that when a policyholder incurs a loss, the size of the loss (in thousands of dollars) is where the probability density function of is where . Suppose that the insurance company just receives a notification that the policyholder had incurred a loss over $2000, what is the probability that the loss exceeds $3000?

**Solution is found below**.

**Solution to Problem 9a**

**More Direct Solution**

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**The original solution**

Note that the support of is (the values with positive probabilities). When , we have . First derive the cdf of and then derive the cdf of . Upon differentiation, the density function of is obtained, which is then used to compute the variance. The following shows the derivations.

.

**Answer to Problem 9b**

Hint. It is not necessary to derive the distribution of before computing the probabilities concerning . For example, .

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Attn Readers: It might be a helpful hint to know that the inverse is in the form of a familiar distribution. This is not specified in the solutions, but simplifies the problem immensely!

Katy Jean. Thanks for all your comments. There is a more direct way to work this problem. Please the direct solution and the hint I added.

I was computing the inverse, seeing it as the Gamma, and then using the CDF and variance formulas for Gamma (which I have memorized), which seemed like a shortcut. Your direct solution might be even easier. Personally, I avoid actually evaluating integrals (even easy ones) under the pressure of a test– it’s like doing long division with my 4th grade teacher watching over my shoulder– too many chances for silly mistakes.

I sympathize with your desire to avoid evaluating integrals in a test. The good thing is that Exam P problems likely do not have monster integrals. If an integral in exam P turns out to be sucking up a lot of exam time, you probably do not look at the problem in a right way. So one job in exam P preparation is to learn various thought processes that can take you to the heart of problems. The way I see it, some integrals are necessary in Exam P problems. For example, integrals involving powers of (such as ). Other examples are integrals involving the density functions of exponential distributions. Another example is the kind of integrals where you can algebraically manipulate to look like the density of a Gamma distribution.