Solutions - 5250-03

(1) Compute $P[B]=P[X>5]$:

$$\begin{array}{c}P[X>5]={\int }_5^{10}\frac{50}{6}{x}^{-3}dx\\ \\ \gg \gg {\frac{-25}{6}{x}^{-2}|}_5^{10}\\ \\ \gg \gg \frac{-25}{6}({10}^{-2}-5^{-2})\gg \gg 0.125\end{array}$$

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(2) Compute the conditional density $f_{X|B}$:

$$f_{X|B}(x)=\frac{\frac{50}{6}{x}^{-3}}{0.125},5<x<10$$

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(3) Compute $P[X>7|X>5]$:

$$\begin{array}{c}P[X>7|X>5]={\int }_7^{10}\frac{\frac{50}{6}{x}^{-3}}{0.125}dx\\ \\ \gg \gg \frac{1}{0.125}\cdot \frac{50}{6}{\left[\frac{-1}{2}{x}^{-2}\right]}_7^{10}\\ \\ \gg \gg \frac{-100}{3}({10}^{-2}-7^{-2})\gg \gg \approx 0.3469\end{array}$$