A particle is moving on the unit circle x² + y² = 1 at the constant angular speed > 0, starting at the initial position (1,0). Here, by angular speed w, it means it takes t = 2 to return to the initial position. The particle will stop at the random time T, which has distribution T~ exp(1). (a) Find the probability that the particle stops at the k-th quadrant, for k = 1, 2, 3, 4. Your answer depends on w. Decide which quadrant has the highest probability. (b) Analyze the limiting probability from (a) as the angular speed w→ ∞. (b) Let X be the x-coordinate of the particle's final position. Find the expected value and variance of X. (c) Fix w = 1, and let X be the x-coordinate of the particle's final position. Find E[X] [xl
A particle is moving on the unit circle x² + y² = 1 at the constant angular speed > 0, starting at the initial position (1,0). Here, by angular speed w, it means it takes t = 2 to return to the initial position. The particle will stop at the random time T, which has distribution T~ exp(1). (a) Find the probability that the particle stops at the k-th quadrant, for k = 1, 2, 3, 4. Your answer depends on w. Decide which quadrant has the highest probability. (b) Analyze the limiting probability from (a) as the angular speed w→ ∞. (b) Let X be the x-coordinate of the particle's final position. Find the expected value and variance of X. (c) Fix w = 1, and let X be the x-coordinate of the particle's final position. Find E[X] [xl
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 particle is moving on the unit circle x² + y² = 1 at the constant angular
speed > 0, starting at the initial position (1,0). Here, by angular speed w, it means it
takes t = 2 to return to the initial position. The particle will stop at the random time T,
which has distribution T ~ exp(1).
W
=
1, 2, 3, 4.
(a) Find the probability that the particle stops at the k-th quadrant, for k
Your answer depends on w. Decide which quadrant has the highest probability.
(b) Analyze the limiting probability from (a) as the angular speed w → ∞.
(b) Let X be the x-coordinate of the particle's final position. Find the expected value
and variance of X.
(c) Fix w = 1, and let X be the x-coordinate of the particle's final position. Find E[X]
and Var[X].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd94c6fe5-0a79-4b34-8240-29323f6fd175%2Fce3ff472-5d87-42b1-817b-ff2b44557e21%2Ff3ps2bk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A particle is moving on the unit circle x² + y² = 1 at the constant angular
speed > 0, starting at the initial position (1,0). Here, by angular speed w, it means it
takes t = 2 to return to the initial position. The particle will stop at the random time T,
which has distribution T ~ exp(1).
W
=
1, 2, 3, 4.
(a) Find the probability that the particle stops at the k-th quadrant, for k
Your answer depends on w. Decide which quadrant has the highest probability.
(b) Analyze the limiting probability from (a) as the angular speed w → ∞.
(b) Let X be the x-coordinate of the particle's final position. Find the expected value
and variance of X.
(c) Fix w = 1, and let X be the x-coordinate of the particle's final position. Find E[X]
and Var[X].
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