A particle starts from the origin and moves in a certain direction for a distance 1₁. Then it changes its direction many times, making a random walk: for a distance /2, then for a distance 13, and so forth. The trajectory of the wandering particle consists thus of segments of lengths 1₁, 12,..., , the direction of each being determined by the angle ak made with the Ox-axis. These angles are uniformly distributed in the interval (0, 2), and they are independent. Find the characteristic function of the coordinate X of the end- point of the trajectory and the corresponding probability density.
A particle starts from the origin and moves in a certain direction for a distance 1₁. Then it changes its direction many times, making a random walk: for a distance /2, then for a distance 13, and so forth. The trajectory of the wandering particle consists thus of segments of lengths 1₁, 12,..., , the direction of each being determined by the angle ak made with the Ox-axis. These angles are uniformly distributed in the interval (0, 2), and they are independent. Find the characteristic function of the coordinate X of the end- point of the trajectory and the corresponding probability density.
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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Transcribed Image Text:A particle starts from the origin and moves in a certain
direction for a distance 1₁. Then it changes its direction many times, making a
random walk: for a distance /2, then for a distance 13, and so forth. The trajectory
of the wandering particle consists thus of segments of lengths 1₁, 12,..., , the
direction of each being determined by the angle ak made with the Ox-axis.
These angles are uniformly distributed in the interval (0, 2), and they are
independent. Find the characteristic function of the coordinate X of the end-
point of the trajectory and the corresponding probability density.
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