91Ó°ÊÓ

An unknown number, unknown value, or unknown quantity is represented by a variable, which is an alphabet or word. In the context of algebraic expressions or algebra, the variables are particularly useful.

The product of\({\rm{\pi }}\)and the radius squared is the area of a circle, with the radius dispersed according to the random variable R:

\({\rm{A = \pi }}{{\rm{r}}^{\rm{2}}}\)

Take each side of the equation's expected value:

\(\begin{array}{c}{\rm{E(A) = E}}\left( {{\rm{\pi }}{{\rm{r}}^{\rm{2}}}} \right)\\{\rm{ = \pi E}}\left( {{{\rm{r}}^{\rm{2}}}} \right)\end{array}\)

The integral of the product of each possibility\({\rm{x}}\)with its probability\({\rm{P(x)}}\)is the expected value (or mean)\({\rm{\mu }}\):

\(\begin{aligned}{\rm{E}}\left( {{{\rm{r}}^{\rm{2}}}} \right) &= \int_{{\rm{ - }}\infty }^{{\rm{ + }}\infty } {{{\rm{r}}^{\rm{2}}}} {\rm{f(r)dr}}\\ &= \frac{{\rm{3}}}{{\rm{4}}}\int_{\rm{9}}^{{\rm{11}}} {{{\rm{r}}^{\rm{2}}}} \left( {{\rm{1 - (10 - r}}{{\rm{)}}^{\rm{2}}}} \right){\rm{dr}}\\ &= \frac{{\rm{3}}}{{\rm{4}}}\int_{\rm{9}}^{{\rm{11}}} {{{\rm{r}}^{\rm{2}}}} \left( {{\rm{1 - 100 + 20r - }}{{\rm{r}}^{\rm{2}}}} \right){\rm{dr}}\\ &= \frac{{\rm{3}}}{{\rm{4}}}\int_{\rm{9}}^{{\rm{11}}} {\rm{ - }} {\rm{99}}{{\rm{r}}^{\rm{2}}}{\rm{ + 20}}{{\rm{r}}^{\rm{3}}}{\rm{ - }}{{\rm{r}}^{\rm{4}}}{\rm{dr}}\end{aligned}\)

\(\begin{aligned} & = \left. {\frac{{\rm{3}}}{{\rm{4}}}\left( {{\rm{ - 33}}{{\rm{r}}^{\rm{3}}}{\rm{ + 5}}{{\rm{r}}^{\rm{4}}}{\rm{ - }}\frac{{{{\rm{r}}^{\rm{5}}}}}{{\rm{5}}}} \right)} \right|_{\rm{9}}^{{\rm{11}}}\\ &= \frac{{\rm{3}}}{{\rm{4}}}{\rm{ \times }}\frac{{{\rm{668}}}}{{\rm{5}}}\\ &= \frac{{{\rm{501}}}}{{\rm{5}}}\end{aligned}\)

Determine the area's expected value:

\(\begin{aligned}{\rm{E(A) = E}}\left( {{\rm{\pi }}{{\rm{r}}^{\rm{2}}}} \right)\\ &= \pi E\left( {{{\rm{r}}^{\rm{2}}}} \right)\\ &= \frac{{{\rm{501\pi }}}}{{\rm{5}}}\\ \approx {\rm{314}}{\rm{.79}}\end{aligned}\)

Therefore, the value is \( \approx {\rm{314}}{\rm{.79}}\).