An ecologist wishes to mark off a circular sampling region having radius 9 m. However, the radius of the resulting region is actually a random variable m²-f¾1- 1-(9-7)²] 8 ≤rs 10 0 otherwise What is the expected area of the resulting circular region? (Round your answer to two decimal places.) with the following pdf.
An ecologist wishes to mark off a circular sampling region having radius 9 m. However, the radius of the resulting region is actually a random variable m²-f¾1- 1-(9-7)²] 8 ≤rs 10 0 otherwise What is the expected area of the resulting circular region? (Round your answer to two decimal places.) with the following pdf.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![**Expected Area of a Circular Sampling Region**
An ecologist wishes to mark off a circular sampling region having a radius of \( 9 \) m. However, the radius of the resulting region is actually a random variable \( R \) with the following probability density function (pdf):
\[ f(r) =
\begin{cases}
\frac{3}{4} \left[ 1 - (9 - r)^2 \right] & 8 \leq r \leq 10 \\
0 & \text{otherwise}
\end{cases} \]
Given this, we are to determine the expected area of the resulting circular region. (Round your answer to two decimal places.)
\[ \text{Expected Area} = \boxed{} \, \text{m}^2 \]
To find the expected area, we must first calculate the expected value \( E[R] \) of the radius \( R \). Afterwards, we can use the formula for the area of a circle, \( A = \pi R^2 \), to determine the expected area.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda5a8041-aeed-45a6-a7fc-c4165c2d9a1f%2Fb7773590-3475-4da8-ba2a-78eb1ad6b4f7%2Fir3m4rt_processed.png&w=3840&q=75)
Transcribed Image Text:**Expected Area of a Circular Sampling Region**
An ecologist wishes to mark off a circular sampling region having a radius of \( 9 \) m. However, the radius of the resulting region is actually a random variable \( R \) with the following probability density function (pdf):
\[ f(r) =
\begin{cases}
\frac{3}{4} \left[ 1 - (9 - r)^2 \right] & 8 \leq r \leq 10 \\
0 & \text{otherwise}
\end{cases} \]
Given this, we are to determine the expected area of the resulting circular region. (Round your answer to two decimal places.)
\[ \text{Expected Area} = \boxed{} \, \text{m}^2 \]
To find the expected area, we must first calculate the expected value \( E[R] \) of the radius \( R \). Afterwards, we can use the formula for the area of a circle, \( A = \pi R^2 \), to determine the expected area.
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