Exam P Practice Problem 9

Problem 9a
Suppose that when a policyholder incurs a loss, the size of the loss (in thousands of dollars) is $Y=X^{-1}$ where the probability density function of $X$ is $f(x)=2.5 x^{1.5}$ where $0. Find the variance of the size of loss.

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Problem 9b
Suppose that when a policyholder incurs a loss, the size of the loss (in thousands of dollars) is $Y=X^{-1}$ where the probability density function of $X$ is $f(x)=3 x^{2}$ where $0. 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?

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Solution is found below.

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Solution to Problem 9a

More Direct Solution

$\displaystyle E(Y)=E(X^{-1})=\int_0^1 x^{-1} \ 2.5 x^{1.5} \ dx =\frac{5}{3}$

$\displaystyle E(Y^2)=E(X^{-2})=\int_0^1 x^{-2} \ 2.5 x^{1.5} \ dx =5$

$\displaystyle Var(Y)=5- \biggl(\frac{5}{3}\biggr)^2=\frac{20}{9}$

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The original solution

Note that the support of $Y$ is $y>1$ (the values with positive probabilities). When $0, we have $y>1$. First derive the cdf of $X$ and then derive the cdf of $Y=X^{-1}$. Upon differentiation, the density function of $Y$ is obtained, which is then used to compute the variance. The following shows the derivations.

$\displaystyle F_X(x)=\left\{\begin{matrix}0& \ \ \ \ \ \ x \le 0 \\{\text{ }}& \\{x^{2.5}}&\ \ \ \ \ \ 0 < x <1 \\{\text{ }}& \\{1}&\ \ \ \ \ \ x \ge 1 \end{matrix}\right.$

\displaystyle \begin{aligned}F_Y(y)&=P(Y \le y) \\&=P(X^{-1} \le y) \\&=P(X>y^{-1}) \\&=1-F_X(y^{-1}) \\&=1-y^{-2.5},y>1 \end{aligned}

$\displaystyle f_Y(y)=2.5 y^{-3.5}$.

$\displaystyle E(Y)=\int \limits_{1} ^\infty 2.5 y^{-2.5} \ dy=\frac{5}{3}$

$\displaystyle E(Y^2)=\int \limits_{1} ^\infty 2.5 y^{-1.5} \ dy=5$

$\displaystyle Var(Y)=5-\frac{25}{9}=\frac{20}{9}$

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$\displaystyle \frac{8}{27}=0.2963$

Hint. It is not necessary to derive the distribution of $Y$ before computing the probabilities concerning $Y$. For example, $P(Y > 2)=P(X^{-1} > 2)=P(X < 2^{-1})$.

2. 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 $x$ (such as $\int_a^b x^t \ dx$). 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.