6. This question is a simple exercise in changing variables. A gun fires at random in the angular range −π/2 < 0 < π/2 towards a wall a distance away. If y is the coordinate along the wall, show that 1 g(y)dy = 1 dy π 1 + (y/l)² 1 This, as we saw in the lectures, is the Cauchy distribution. Assuming that the mean should be zero from symmetry considerations, try to find the standard deviation; what problem do you have? Truncate the distribution at a distance |y| = L (not the same as 1) either side of the peak. Calculate the new normalisation constant, and find the standard deviation.
6. This question is a simple exercise in changing variables. A gun fires at random in the angular range −π/2 < 0 < π/2 towards a wall a distance away. If y is the coordinate along the wall, show that 1 g(y)dy = 1 dy π 1 + (y/l)² 1 This, as we saw in the lectures, is the Cauchy distribution. Assuming that the mean should be zero from symmetry considerations, try to find the standard deviation; what problem do you have? Truncate the distribution at a distance |y| = L (not the same as 1) either side of the peak. Calculate the new normalisation constant, and find the standard deviation.
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:6. This question is a simple exercise in changing variables. A gun fires at random in the angular
range −π/2 < 0 < π/2 towards a wall a distance away. If y is the coordinate along the wall,
show that
1
g(y)dy
=
1 dy
π 1 + (y/l)² 1
This, as we saw in the lectures, is the Cauchy distribution. Assuming that the mean should be
zero from symmetry considerations, try to find the standard deviation; what problem do you
have? Truncate the distribution at a distance |y| = L (not the same as 1) either side of the
peak. Calculate the new normalisation constant, and find the standard deviation.
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