Partnership for Integration of Computation into Undergraduate Physics

1. Produce a large set of numbers that obey a normal distribution with a mean of 5.4 and a standard deviation of 0.2. Many computer programming languages have a weighted random number built in. You can also build your own using the Box Muller transformation that takes two uniformly distributed random numbers between 0 and 1 and returns two normally distributed random numbers with a mean of zero and a standard deviation of 1. These can then be transformed to match the requested mean and standard deviation. Plot a histogram of the large data set, confirming that the peak and width are what you expected. 2. Determine the histogram for the speed example above by using the calculus error propagation approach. a) Assuming that all values are distributed according to a normal distribution, the calculus approach is given by $$\sigma_f=\sqrt{\left(\frac{\partial f}{\partial x}\sigma_x\right)^2+\left(\frac{\partial f}{\partial y}\sigma_y\right)^2}$$ Show that in the case of the speed calculation, this becomes : $$\sigma_v=\frac{x}{t}\sqrt{\left(\frac{\sigma_x}{x}\right)^2+\left(\frac{\sigma_y}{y}\right)^2}$$ b) Using Eq. 2, determine the error for the speed example from Exercise 1. 3. Now use the Monte Carlo method. Generate several hundreds or thousands of values for both position and time according to their respective distributions. Calculate the speed that corresponds to each of these values, and plot a histogram of speeds. Compare with the results of exercise 2. 3. Produce several (hundreds or thousands) of both position and time estimates according to their respective distributions and calculate their associated speeds. Then plot the histogram of those speeds and compare with (2). 4. Determine the mean, median, and standard deviation of the histrogram for (3) and compare with 2. 5. Extend the Monte Carlo approach to lab data of your own.