Solutions - 5120-02

(1) State the PDF of $X$ and the payout function $g(X)$:

Since repair cost is uniformly distributed, $f_X(x)=\frac{1}{1400}$ for $100\le x\le 1500$.

The payout with a $500 deductible is:

$$g(x)=\{\begin{array}{cc}0& x\le 500\\ x-500& x>500\end{array}$$

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(2) Compute $E[g(X)]$:

$$\begin{array}{c}E[g(X)]={\int }_{-\infty }^{\infty }g(x)f_X(x)dx\\ \\ \gg \gg \frac{1}{1400}{\int }_{500}^{1500}(x-500)dx\\ \\ \gg \gg \frac{1}{1400}{[\frac{x^2}{2}-500x]|}_{500}^{1500}\gg \gg \approx \$357.14\end{array}$$