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Chapter 4: Q21E (page 156)

An ecologist wishes to mark off a circular sampling region having radius \({\rm{10\;m}}\). However, the radius of the resulting region is actually a random variable \({\rm{R}}\) with pdf

\({\rm{f(r) = }}\left\{ {\begin{array}{*{20}{c}}{\frac{{\rm{3}}}{{\rm{4}}}\left( {{\rm{1 - (10 - r}}{{\rm{)}}^{\rm{2}}}} \right)}&{{\rm{9}} \le {\rm{r}} \le {\rm{11}}}\\{\rm{0}}&{{\rm{ otherwise }}}\end{array}} \right.\)

What is the expected area of the resulting circular region?

Short Answer

Expert verified

The value is \( \approx {\rm{314}}{\rm{.79}}\).

Step by step solution

01

Define variable

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.

02

Explanation

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 \( \approx {\rm{314}}{\rm{.79}}\).

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Most popular questions from this chapter

Let X denote the distance \({\rm{(m)}}\) that an animal moves from its birth site to the first territorial vacancy it encounters. Suppose that for banner-tailed kangaroo rats, X has an exponential distribution with parameter \({\rm{\lambda = }}{\rm{.01386}}\) (as suggested in the article "Competition and Dispersal from Multiple Nests," Ecology, 1997: 873-883).

a. What is the probability that the distance is at most \({\rm{100\;m}}\)? At most \({\rm{200\;m}}\) ? Between 100 and\({\rm{200\;m}}\)?

b. What is the probability that distance exceeds the mean distance by more than 2 standard deviations?

c. What is the value of the median distance?

A \(12\)-in. bar that is clamped at both ends is to be subjected to an increasing amount of stress until it snaps. Let Y = the distance from the left end at which the break occurs. Suppose Y has pdf

\(f\left( y \right) = \left\{ {\begin{array}{*{20}{c}}{\left( {\frac{1}{{24}}} \right)y\left( {1 - \frac{y}{{12}}} \right)\,\,\,\,\,0 \le y \le 12}\\{0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,otherwise}\end{array}} \right.\)

Compute the following: a. The cdf of Y, and graph it. b.\(P\left( {Y \le 4} \right), P\left( {Y > 6} \right)\), and\(P\left( {4 \le Y \le 6} \right)\)c. E(Y), E(Y2 ), and V(Y) d. The probability that the breakpoint occurs more than \(2\;\)in. from the expected breakpoint. e. The expected length of the shorter segment when the break occurs.

A family of pdf鈥檚 that has been used to approximate the distribution of income, city population size, and size of firms is the Pareto family. The family has two parameters, \({\rm{k}}\) and \({\rm{\theta }}\), both\({\rm{ > 0}}\), and the pdf is

\({\rm{f(x;\theta ) = \{ }}\begin{array}{*{20}{c}}{\frac{{{\rm{k}} \cdot {{\rm{\theta }}^{\rm{k}}}}}{{{{\rm{x}}^{{\rm{k + 1}}}}}}}&{{\rm{x}} \ge {\rm{\theta }}}\\{\rm{0}}&{{\rm{x < \theta }}}\end{array}\)

a. Sketch the graph of \({\rm{f(x;\theta )}}\).

b. Verify that the total area under the graph equals \({\rm{1}}\).

c. If the rv \({\rm{X}}\) has pdf \({\rm{f(x;\theta )}}\), for any fixed \({\rm{b > \theta }}\), obtain an expression for \({\rm{P(X}} \le {\rm{b)}}\).

d. For \({\rm{\theta < a < b}}\) obtain an expression for the probability \({\rm{P(a}} \le {\rm{X}} \le {\rm{b)}}\).

If bolt thread length is normally distributed, what is the probability that the thread length of a randomly selected bolt is a. Within \(1.5\)SDs of its mean value? b. Farther than \(2.5\)SDs from its mean value? c. Between \(1 and 2\)SDs from its mean value?

a. The event \(\left\{ {{X^2} \le y} \right\}\)is equivalent to what event involvingXitself?

b. If \(X\)has a standard normal distribution, use part (a) to write the integral that equals \(P\left( {{X^2} \le y} \right)\). Then differentiate this with respect to \(y\)to obtain the pdf of \({{\rm{X}}^{\rm{2}}}\) (the square of a \({\rm{N(0,1)}}\)variable). Finally, show that \({{\rm{X}}^{\rm{2}}}\)has a chi-squared distribution with \(\nu = 1\) df (see (4.10)). (Hint: Use the following identity.)

\(\frac{d}{{dy}}\left\{ {\int_{a(y)}^{b(y)} f (x)dx} \right\} = f(b(y)) \cdot {b^\prime }(y) - f(a(y)) \cdot {a^\prime }(y)\)
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