2.0.2 Problem 2: Error propagationa) You are given the measurements of two sides of a rectangle A and B with the associate errorsA and B, respectively. Assuming A and B are uncorrelated, calculate the error in the: >(i) sum A + B (ii) difference A - B (iii) the perimeter 2A + 2B (iv) the area A x B (v) theaspect ratio A/Bb) In the muon lifetime experiment, we obtain a histogram for the recorded decays as function ofthe time after the muon enters the detector. After binning the decays into discrete time bins,we expect the distribution of decays (histogram) to be described by an exponential functionof time. Rather than fitting the data directly with an exponential function, it is sometimesmore convenient to plot the logarithm of the decays in a bin (y;) as a function of time (x₁) andthen fit a straight line to it. Each data point of the histogram (x, y) has a statistical error,oi √, associated with it. What happens to these errors when the semi-log histogram(x, log10 yi) is plotted? Assume that all values of y, are >> 1.=c) In a separate experiment, you find that log₁0 Eo = 1.5±0.5 (at 68% confidence level, CL).What is the value of Eo and the experimental bounds at 68% CL? (Note that 0.5 is not smallcompared to 1.5).

Answered: You are given the measurements of two…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

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2.0.2 Problem 2: Error propagation
a) You are given the measurements of two sides of a rectangle A and B with the associate errors
A and B, respectively. Assuming A and B are uncorrelated, calculate the error in the: >
(i) sum A + B (ii) difference A - B (iii) the perimeter 2A + 2B (iv) the area A x B (v) the
aspect ratio A/B
b) In the muon lifetime experiment, we obtain a histogram for the recorded decays as function of
the time after the muon enters the detector. After binning the decays into discrete time bins,
we expect the distribution of decays (histogram) to be described by an exponential function
of time. Rather than fitting the data directly with an exponential function, it is sometimes
more convenient to plot the logarithm of the decays in a bin (y;) as a function of time (x₁) and
then fit a straight line to it. Each data point of the histogram (x, y) has a statistical error,
oi √, associated with it. What happens to these errors when the semi-log histogram
(x, log10 yi) is plotted? Assume that all values of y, are >> 1.
=
c) In a separate experiment, you find that log₁0 Eo = 1.5±0.5 (at 68% confidence level, CL).
What is the value of Eo and the experimental bounds at 68% CL? (Note that 0.5 is not small
compared to 1.5).

Expert Solution

Step 1

Note: As per the guidelines, in case of multiple questions, answer to first question will be provided. Kindly post other questions separately.

General formula for Error propagation:

If there are n measurements $x_{1}, x_{2}, . . . ., x_{n}$ with the uncertainties $σ_{1}, σ_{2}, . . . ., σ_{n}$ , and a function of the measurements $f (x_{1}, . . ., x_{n})$ is known, then the uncertainty in f is given by:

$σ_{f} = \sqrt{{(\frac{\partial f}{\partial x_{1}} σ_{1})}^{2} + {(\frac{\partial f}{\partial x_{2}} σ_{2})}^{2} + . . . . + {(\frac{\partial f}{\partial x_{n}} σ_{n})}^{2}}$