Exercise 19.7

Exercise 19.7#

Question 1#

Consider the following relation to determine a variable \(Y\) from variables \(X_i\) with known quantities:

\[ Y = X_1 + X_2 + X_3 + X_4 \]

The \(X_i\) variables all have uniform dispersion, with measured quantities:

  • \(x_1 = 10.0\), \(u(x_1) = 1.0\)

  • \(x_2 = 20.0\), \(u(x_2) = 2.0\)

  • \(x_3 = 30.0\), \(u(x_3) = 3.0\)

  • \(x_4 = 40.0\), \(u(x_4) = 4.0\)

Use \(M = 10^6\) samples to find the best approximation of \(Y\): \(y\) and the standard uncertainty \(u(y)\) using Monte Carlo uncertainty propagation.

As part of this process, plot a histogram of your \(y_r\) samples. Visualize how well a normal distribution fits by plotting a Gaussian over the histogram (use the mean and experimental standard deviation of the \(y_r\) to determine the parameters for the plot). Even if it’s not an exact fit, you may assume the \(y_r\) are sufficiently normally distributed to find the best approximation and standard uncertainty.

Compare your results from the Monte Carlo methods to the results from the combined uncertainty equation (28), for which you can calculate the best approximation of \(Y\) directly from the \(x_i\) quantities and the standard uncertainty as:

\[ u(y) = \sqrt{u(x_1)^2 + u(x_2)^2 + u(x_3)^2 + u(x_4)^2} \]

Do the results from the Monte Carlo method and the direct calculation agree given a level of confidence of \(68.27\%\)? (You may assume that the different \(y\) results as well as their differences are normally distributed)

Question 2#

In Question 1 you were told to approximate the distribution of \(y_r\) values as normal, however you may have found that this doesn’t appear to be a good approximation. A metric that can be used to measure how well a data-set is characterised by a normal distribution is to approximate it’s excess kurtosis [UMC-Ex1]; with a perfectly normally distributed data-set having an excess kurtosis of 0. A (biased) estimator for the excess kurtosis of a data-set is defined as:

\[ g_2 \equiv \frac{\tfrac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^4}{\left[\tfrac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2\right]^2} - 3 \]

Consider the generic relation used to determine a variable \(Y\) from variables \(X_i\) with known quantities:

\[ Y = \sum_{i=1}^N X_i \]

where \(X_i\) has measured quantities with best approximations and uncertainties:

\[ x_i = 10~i \quad u(x_i) = i \]

Compare the distributions of \(y_r\) and its excess kurtosis for a range of \(N\) values from 1 to 7 and for \(X_i\) with uniform, triangular and normally distributed dispersion. Use two sets of subplots:

  • Histograms of the distributions of \(y_r\) for different values of \(N\) and different distributions for the \(X_i\)

  • Line plots of the excess kurtosis vs \(N\) for the different distributions for the \(X_i\)

What trend do you notice as \(N\) increases? Are the distributions of \(y_r\) becoming more normal?

References#

[UMC-Ex1]

Wikipedia contributors. Kurtosis — Wikipedia, the free encyclopedia. [Online; accessed 18-May-2026]. URL: https://en.wikipedia.org/wiki/Kurtosis.

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