Queen Dido, the founder of Carthage, was told that she could have as much land for Carthage as could be encircled with a Bull's hide. https://en.wikipedia.org/wiki/Dido Queen Dido cleverly made a rope from the Bull's hide and used the rope to form a semicircle alongside the sea. A semicircle maximizes the area encircled assuming that the coast line is a straight line. It is non-trivial to
Queen Dido, the founder of Carthage, was told that she could have as much land for Carthage as could be encircled with a Bull's hide. https://en.wikipedia.org/wiki/Dido Queen Dido cleverly made a rope from the Bull's hide and used the rope to form a semicircle alongside the sea. A semicircle maximizes the area encircled assuming that the coast line is a straight line. It is non-trivial to
A First Course in Probability (10th Edition)
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Chapter1: Combinatorial Analysis
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Queen Dido cleverly made a rope from the Bull’s hide and used the rope to form a semicircle alongside the sea. A semicircle maximizes the area encircled assuming that the coast line is a straight line. It is non-trivial to show that this shape maximizes the area.
\[http://liberzon.csl.illinois.edu/teaching/cvoc/node21.html\]
Let \( L \) be the length of the rope in kilometers, and \( A \) be the area encircled by the rope along the coast.
(a) Express \( A \) as a function of \( L \).
(b) Express the mean of \( A \) as a function of the mean \( \mu \) and variance \( \sigma^2 \) of \( L \).
(c) Assume that \( L \) is exponentially distributed with mean \( 1/2 \) km. What is the c.d.f. of \( A \)?
(d) Assume that \( L \) is exponentially distributed with mean \( 1/2 \) km. Find the pmf or pdf of \( L \), whichever is appropriate.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb9e27b5d-0ab8-428e-954d-97d64fc14c61%2F79cd7dea-8222-4e0f-a71e-34b553ccd4e8%2Fdp1o377_processed.png&w=3840&q=75)
Transcribed Image Text:2. Queen Dido, the founder of Carthage, was told that she could have as much land for Carthage as could be encircled with a Bull’s hide.
[Source](https://en.wikipedia.org/wiki/Dido)
Queen Dido cleverly made a rope from the Bull’s hide and used the rope to form a semicircle alongside the sea. A semicircle maximizes the area encircled assuming that the coast line is a straight line. It is non-trivial to show that this shape maximizes the area.
\[http://liberzon.csl.illinois.edu/teaching/cvoc/node21.html\]
Let \( L \) be the length of the rope in kilometers, and \( A \) be the area encircled by the rope along the coast.
(a) Express \( A \) as a function of \( L \).
(b) Express the mean of \( A \) as a function of the mean \( \mu \) and variance \( \sigma^2 \) of \( L \).
(c) Assume that \( L \) is exponentially distributed with mean \( 1/2 \) km. What is the c.d.f. of \( A \)?
(d) Assume that \( L \) is exponentially distributed with mean \( 1/2 \) km. Find the pmf or pdf of \( L \), whichever is appropriate.
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