A chord of a circle is a line segment connecting two points on the circle. To pick a point uniformly in the unit circle means that you are randomly throwing a dart at the unit circle. Pick a point Q uniformly at random in the unit circle, radius 0 < R <1 from the origin, and pick any chord of the circle with that point as the midpoint. Let L be the length of this chord. In the following steps, you will find E[L]. 1. Notice that if we rotate the circle, we can assume that the chord is always vertical with non-negative horizontal coordinate (i.e. assume Q=(R,0) for R > 0). (R,0) Show that the CDF of R is FR(r) = r². (Hint: What is the probability that the radius R of the point Q is less than r?) 2. Find the density of R. 3. Find E[L].
A chord of a circle is a line segment connecting two points on the circle. To pick a point uniformly in the unit circle means that you are randomly throwing a dart at the unit circle. Pick a point Q uniformly at random in the unit circle, radius 0 < R <1 from the origin, and pick any chord of the circle with that point as the midpoint. Let L be the length of this chord. In the following steps, you will find E[L]. 1. Notice that if we rotate the circle, we can assume that the chord is always vertical with non-negative horizontal coordinate (i.e. assume Q=(R,0) for R > 0). (R,0) Show that the CDF of R is FR(r) = r². (Hint: What is the probability that the radius R of the point Q is less than r?) 2. Find the density of R. 3. Find E[L].
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![A chord of a circle is a line segment connecting two points on the circle. To pick a point uniformly
in the unit circle means that you are randomly throwing a dart at the unit circle. Pick a point
Q uniformly at random in the unit circle, radius 0 < R <1 from the origin, and pick any chord
of the circle with that point as the midpoint. Let L be the length of this chord.
In the following steps, you will find E[L].
1.
Notice that if we rotate the circle, we can assume that the chord is always vertical
with non-negative horizontal coordinate (i.e. assume Q=(R,0) for R > 0).
(R,0)
Show that the CDF of R is FR(r) = r². (Hint: What is the probability that the radius R
of the point Q is less than r?)
2.
Find the density of R.
3.
Find E[L].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F74e614b0-22c1-4100-8389-98bc16ab838d%2F38e605e7-46b1-41f9-bdb0-b73f10b1aa0f%2Fmxz0in_processed.png&w=3840&q=75)
Transcribed Image Text:A chord of a circle is a line segment connecting two points on the circle. To pick a point uniformly
in the unit circle means that you are randomly throwing a dart at the unit circle. Pick a point
Q uniformly at random in the unit circle, radius 0 < R <1 from the origin, and pick any chord
of the circle with that point as the midpoint. Let L be the length of this chord.
In the following steps, you will find E[L].
1.
Notice that if we rotate the circle, we can assume that the chord is always vertical
with non-negative horizontal coordinate (i.e. assume Q=(R,0) for R > 0).
(R,0)
Show that the CDF of R is FR(r) = r². (Hint: What is the probability that the radius R
of the point Q is less than r?)
2.
Find the density of R.
3.
Find E[L].
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